How many questions from Definite Integration appear in JEE?

Definite Integration has appeared in 168 of the last 186 JEE Main and JEE Advanced papers — about 90% of them — contributing 267 questions in total across those papers.

Is Definite Integration an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 90% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Definite Integration questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Definite Integration JEE Previous Year Questions with Answers

Q1·MathematicsNumericalJEE Main 2026

If

\displaystyle\int_{\pi/6}^{\pi/4} \left(\cot\left(x - \dfrac{\pi}{3}\right)\cot\left(x + \dfrac{\pi}{3}\right) + 1\right)dx = \alpha \log_e(\sqrt{3} - 1)

, then

9\alpha^2

is equal to ________.

Q2·MathematicsSingle correctJEE Main 2026

The value of the integral

\displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2+x+1)}}{(\sqrt{x+1})(\sqrt{x^4+x^2+1})}\,dx

is equal to:

(A) $\dfrac{1}{3}\log_e(3 - 2\sqrt{2})$
(B) $\dfrac{2}{3}\log_e(4 + \sqrt{2})$
(C) $\dfrac{2}{3}\log_e(3 + 2\sqrt{2})$
(D) $\dfrac{1}{3}\log_e(1 + 6\sqrt{2})$

Q3·MathematicsSingle correctJEE Main 2026

The value of the integral

\int_{-1}^{1} \left( \frac{x^{3} + |x| + 1}{x^{2} + 2|x| + 1} \right) dx

is equal to :

(A) $3\log_{e} 2$
(B) $2\log_{e} 2$
(C) $5\log_{e} 3$
(D) $3\log_{e} 3$

Q4·MathematicsSingle correctJEE Main 2026

The value of the integral

\int_{-\pi/4}^{\pi/4}\left(\frac{32\cos^4 x}{1+e^{\sin x}}\right)dx

is:

(A)4π + 2
(B)3π + 8
(C)3π + 4
(D)4π + 3

Q5·MathematicsSingle correctJEE Main 2026

Let

A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}

be a singular matrix. Let

f(x) = \int_{0}^{x} (t^{2} + 2t + 3)\, dt

,

x \in [1, \alpha]

. If

M

and

m

are respectively the maximum and the minimum values of

f

in

[1, \alpha]

, then

3(M - m)

is equal to :

(A)64
(B)68
(C)72
(D)76

Q6·MathematicsSingle correctJEE Main 2026

The value of the integral

\int_{\pi/6}^{\pi/3} \left( \frac{4 - \csc^2 x}{\cos^4 x} \right) dx

is:

(A) $\frac{11}{\sqrt{3}}$
(B) $\frac{16}{\sqrt{3}}$
(C) $\frac{32}{3\sqrt{3}}$
(D) $\frac{64}{3\sqrt{3}}$

Q7·MathematicsSingle correctJEE Main 2026

Let

(2^{1-a} + 2^{1+a})

, f(a),

(3^{a} + 3^{-a})

be in A.P. and α be the minimum value of f(a). Then the value of the integral

\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \frac{dx}{(e^{2x} - e^{-2x})}

is :

(A) $\frac{1}{2}\log_e\left(\frac{4}{3}\right)$
(B) $\frac{1}{4}\log_e\left(\frac{4}{3}\right)$
(C) $\frac{1}{2}\log_e\left(\frac{8}{5}\right)$
(D) $\frac{1}{4}\log_e\left(\frac{8}{5}\right)$

Q8·MathematicsSingle correctJEE Main 2026

The value of the integral

\int_0^\infty \frac{\log_e(x)}{x^2 + 4}\,dx

is:

(A) $\frac{\pi \log_e(2)}{2}$
(B) $\frac{\pi \log_e(2)}{4}$
(C) $1 + \pi \log_e(2)$
(D) $2 + \pi \log_e(2)$

Q9·MathematicsSingle correctJEE Main 2026

The integral

\int_{0}^{1}\cot^{-1}(1 + x + x^{2})dx

is equal to:

(A) $2\tan^{-1}2 + \frac{1}{2}\log_{e}\left(\frac{5}{4}\right) + \frac{\pi}{2}$
(B) $2\tan^{-1}2 + \frac{1}{2}\log_{e}\left(\frac{5}{4}\right) - \frac{\pi}{2}$
(C) $2\tan^{-1}2 - \frac{1}{2}\log_{e}\left(\frac{5}{4}\right) + \frac{\pi}{2}$
(D) $2\tan^{-1}2 - \frac{1}{2}\log_{e}\left(\frac{5}{4}\right) - \frac{\pi}{2}$

Q10·MathematicsNumericalJEE Main 2026

Let

f

be a twice differentiable function such that

f(x) = \int_{0}^{x}\tan(t - x)dt - \int_{0}^{x}f(t)\tan t\,dt

,

x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

. Then

f''\left(\frac{\pi}{6}\right) + 12f'\left(-\frac{\pi}{6}\right) + f\left(\frac{\pi}{6}\right)

is equal to ______

Q11·MathematicsSingle correctJEE Main 2026

Let

f

be a real polynomial of degree

n

such that

f(x) = f'(x)f''(x)

, for all

x \in \mathbb{R}

. If

f(0) = 0

, then

36\left(f'(2) + f''(2) + \int_0^2 f(x)\, dx\right)

is equal to:

(A) $42$
(B) $46$
(C) $56$
(D) $66$

Q12·MathematicsSingle correctJEE Main 2026

Let

\int_{-2}^{2}\left(|\sin x| + [x \sin x]\right) dx = 2(3 - \cos 2) + \beta

, where

[\cdot]

is the greatest integer function. Then

\beta \sin\left(\frac{\beta}{2}\right)

equals:

(A) $1$
(B) $2$
(C) $4$
(D) $8$

Q13·MathematicsSingle correctJEE Main 2026

The value of

\int_0^{20\pi} (\sin^4 x + \cos^4 x)\, dx

is equal to:

(A) $\frac{15\pi}{2}$
(B) $25\pi$
(C) $15\pi$
(D) $\frac{25\pi}{2}$

Q14·MathematicsSingle correctJEE Main 2026

Let

[\cdot]

denote the greatest integer function. Then the value of

\int_{0}^{3}\left(\frac{e^{x} + e^{-x}}{[x]!}\right)dx

is :

(A) $e^{2} + e^{3} - \frac{1}{e^{2}} - \frac{1}{e^{3}}$
(B) $\frac{1}{2}\left(e^{2} + e^{3} - \frac{1}{e^{2}} - \frac{1}{e^{3}}\right)$
(C) $e^{2} + e^{3} - \frac{1}{2e^{2}} - \frac{1}{2e^{3}}$
(D) $\frac{1}{2}\left(e^{2} + e^{3}\right) - \frac{1}{e^{2}} - \frac{1}{e^{3}}$

Q15·MathematicsNumericalJEE Main 2026

If

\alpha = \int_{0}^{2\sqrt{3}}\log_{2}\left(x^{2} + 4\right)dx + \int_{2}^{4}\sqrt{2^{x} - 4}\,dx

, then

\alpha^{2}

is equal to ________.

Q16·MathematicsNumericalJEE Main 2026

The value of

\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_{0}^{r} x\,|\sin \pi x\,|dx\right)}\right|\right)

is ____.

Q17·MathematicsSingle correctJEE Main 2026

Let [·] denote the greatest integer function. Then

\displaystyle\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{12(3+[x])}{3+[\sin x]+[\cos x]}\right) dx

is equal to:

(A) $15\pi + 4$
(B) $11\pi + 2$
(C) $13\pi + 1$
(D) $12\pi + 5$

Q18·MathematicsSingle correctJEE Main 2026

Let

f

be a polynomial function such that

f(x^2 + 1) = x^4 + 5x^2 + 2

, for all

x \in \mathbb{R}

. Then

\int\limits_{0}^{3} f(x)\,dx

is equal to

(A) $\dfrac{41}{3}$
(B) $\dfrac{33}{2}$
(C) $\dfrac{27}{2}$
(D) $\dfrac{5}{3}$

Q19·MathematicsNumericalJEE Main 2026

If f(x) satisfies the relation

f(x) = e^{x} + \int\limits_{0}^{1} (y + xe^{x}) f(y)\, dy

, then

e + f(0)

is equal to ________.

Q20·MathematicsNumericalJEE Main 2026

Let a differentiable function

f

satisfy the equation

\int\limits_{0}^{36} f\left(\frac{\mathrm{tx}}{36}\right) \mathrm{dt} = 4\alpha f(\mathrm{x}).

If y =

f

(x) is a standard parabola passing through the points (2, 1) and (–4, β), Then

\beta^{\alpha}

is equal to ______.

Q21·MathematicsNumericalJEE Main 2026

The number of elements in the set

S = \left\{ x : x \in [0,100] \text{ and } \int_{0}^{x} t^2 \sin(x-t)dt = x^2 \right\}

is.....

Q22·MathematicsSingle correctJEE Main 2026

The value of the integral

\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}}

is :

(A) $\frac{\pi}{12}$
(B) $\frac{\pi}{18}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{3}$

Q23·MathematicsNumericalJEE Main 2026

Let [•] be the greatest integer function. If

\alpha = \int_0^{64}\left(x^{1/3} - \left[x^{1/3}\right]\right)dx

, then

\frac{1}{\pi}\int_0^{\alpha\pi}\left(\frac{\sin^2\theta}{\sin^6\theta + \cos^6\theta}\right)d\theta

is equal to ____.

Q24·MathematicsSingle correctJEE Main 2026

The value of

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right)dx

, where

[\bullet]

denotes the greatest integer function, is

(A) $\frac{1}{60}(21\pi - 1)$
(B) $\frac{1}{60}(\pi - 7)$
(C) $\frac{7}{60}(3\pi - 1)$
(D) $\frac{7}{60}(\pi - 3)$

Q25·MathematicsNumericalJEE Main 2026

6\int_{0}^{\pi}\left|(\sin 3x+\sin 2x+\sin x)\right|dx

is equal to……..

Q26·MathematicsNumericalJEE Main 2026

If

\int_{0}^{1} 4\cot^{-1}(1 - 2x + 4x^{2})dx = a\tan^{-1}(2) - b\log_{e}(5)

, where

a, b \in N

, then

(2a + b)

is equal to _______.

Q27·MathematicsSingle correctJEE Main 2026

The value of

\int_{-\pi/6}^{\pi/6}\left( \frac{\pi + 4x^{11}}{1 - \sin(|x| + \pi/6)} \right) dx

is equal to

(A) $2\pi$
(B) $4\pi$
(C) $8\pi$
(D) $6\pi$

Q28·MathematicsNumericalJEE Advanced 2025

If

\alpha = \int_{1/2}^{2} \frac{\tan^{-1}x}{2x^2 - 3x + 2}\,dx

, then the value of

\sqrt{7}\,\tan\left( \frac{2\alpha\sqrt{7}}{\pi} \right)

is ______ (Here, the inverse trigonometric function

\tan^{-1}x

assumes values in

\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)

.)

Q29·MathematicsSingle correctJEE Main 2025

The integral

\displaystyle\int_{-1}^{3/2}\big(\,|\pi^2 x\sin(\pi x)|\,\big)\,dx

is equal to:

(A) $3+2\pi$
(B) $4+\pi$
(C) $1+3\pi$
(D) $2+3\pi$

Q30·MathematicsSingle correctJEE Main 2025

Let

f(x)

be a positive function and

I_1=\displaystyle\int_{-1/2}^{1}2x\,f\big(2x(1-2x)\big)\,dx

and

I_2=\displaystyle\int_{-1}^{2}f\big(x(1-x)\big)\,dx

. Then the value of

\dfrac{I_2}{I_1}

is equal to ___

(A) $9$
(B) $6$
(C) $12$
(D) $4$

Q31·MathematicsSingle correctJEE Main 2025

The integral

\displaystyle\int_0^\pi\dfrac{(x+3)\sin x}{1+3\cos^2x}\,dx

is equal to:

(A) $\dfrac{\pi}{\sqrt{3}}(\pi+1)$
(B) $\dfrac{\pi}{\sqrt{3}}(\pi+2)$
(C) $\dfrac{\pi}{3\sqrt{3}}(\pi+6)$
(D) $\dfrac{\pi}{2\sqrt{3}}(\pi+4)$

Q32·MathematicsSingle correctJEE Main 2025

Let

f(x)+2f\left(\dfrac{1}{x}\right)=x^2+5

and

2g(x)-3g\left(\dfrac{1}{2}\right)=x

,

x>0

. If

\alpha=\displaystyle\int_1^2 f(x)\,dx

, and

\beta=\displaystyle\int_1^2 g(x)\,dx

, then the value of

9\alpha+\beta

is:

(A)1
(B)0
(C)10
(D)11

Q33·MathematicsSingle correctJEE Main 2025

The value of

\displaystyle\int_{-1}^{1}\dfrac{\left(1+\sqrt{|x|-x}\right)e^x+\left(\sqrt{|x|-x}\right)e^{-x}}{e^x+e^{-x}}\,dx

is equal to:

(A) $3-\dfrac{2\sqrt{2}}{3}$
(B) $2+\dfrac{2\sqrt{2}}{3}$
(C) $1-\dfrac{2\sqrt{2}}{3}$
(D) $1+\dfrac{2\sqrt{2}}{3}$

Q34·MathematicsSingle correctJEE Main 2025

Let the domain of the function

f(x)=\log_2\log_4\log_6(3+4x-x^2)

be

(a,b)

. If

\displaystyle\int_0^{\,b-a}[x^2]\,dx=p-\sqrt q-\sqrt r

,

p,q,r\in\mathbb{N}

,

\gcd(p,q,r)=1

, where

[\cdot]

is the greatest integer function, then

p+q+r

is equal to:

(A)10
(B)8
(C)11
(D)9

Q35·MathematicsSingle correctJEE Main 2025

The integral

\displaystyle\int_0^{\pi/2}\dfrac{8x\,dx}{4\cos^2 x+\sin^2 x}

is equal to:

(A) $2\pi^2$
(B) $4\pi^2$
(C) $\pi^2$
(D) $\dfrac{3\pi^2}{2}$

Q36·MathematicsSingle correctJEE Main 2025

Let

f:[1,\infty)\to[2,\infty)

be a differentiable function. If

10\displaystyle\int_1^x f(t)\,dt = 5x\,f(x) - x^5 - 9

for all

x\ge 1

, then the value of

f(3)

is:

(A)18
(B)32
(C)22
(D)26

Q37·MathematicsIntegerJEE Main 2025

Let

[\cdot]

denote the greatest integer function. If

\displaystyle\int_0^{e^2}\left[\dfrac{1}{e^{x-1}}\right]dx=\alpha-\log_e 2

, then

\alpha^3

is equal to ______.

Q38·MathematicsSingle correctJEE Main 2025

4\displaystyle\int_0^1\left(\dfrac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}}\right)dx-3\log_e(\sqrt3)

is equal to:

(A) $2+\sqrt2+\log_e(1+\sqrt2)$
(B) $2-\sqrt2-\log_e(1+\sqrt2)$
(C) $2+\sqrt2-\log_e(1+\sqrt2)$
(D) $2-\sqrt2+\log_e(1+\sqrt2)$

Q39·MathematicsSingle correctJEE Main 2025

Let

(a,b)

be the point of intersection of the curve

x^2=2y

and the straight line

y-2x-6=0

in the second quadrant. Then the integral

I=\displaystyle\int_a^b \dfrac{9x^2}{1+5^x}\,dx

is equal to:

(A)24
(B)27
(C)18
(D)21

Q40·MathematicsSingle correctJEE Main 2025

Let

f'(x)=\displaystyle\int_0^x t(t^2-9t+20)\,dt

,

1\le x\le 5

. If the range of f is

[\alpha,\beta]

, then

4(\alpha+\beta)

equals:

(A)157
(B)253
(C)125
(D)154

Q41·MathematicsSingle correctJEE Main 2025

The integral

80\displaystyle\int_0^{\pi/2}\left(\dfrac{\sin\theta+\cos\theta}{9+16\sin 2\theta}\right)d\theta

is equal to:

(A) $3\log_e 4$
(B) $6\log_e 4$
(C) $4\log_e 3$
(D) $2\log_e 3$

Q42·MathematicsIntegerJEE Main 2025

Let

f:(0,\infty)\to R

be a twice differentiable function. If for some

a\ne 0

,

\displaystyle\int_0^1 f(\lambda x)\,d\lambda=a\,f(x)

,

f(1)=1

and

f(16)=\dfrac{1}{8}

, then

16-f'\left(\dfrac{1}{16}\right)

is equal to ______.

Q43·MathematicsIntegerJEE Main 2025

If

24\displaystyle\int_0^{\pi/4}\left(\sin\left|4x-\dfrac{\pi}{12}\right|+[2\sin x]\right)dx=2\pi+\alpha

, where

[\cdot]

denotes the greatest integer function, then

\alpha

is equal to ______.

Q44·MathematicsSingle correctJEE Main 2025

Let

f:R\to R

be a twice differentiable function such that

f(2)=1

. If

F(x)=x\,f(x)

for all

x\in R

,

\int_0^2 x\,F'(x)dx=6

and

\int_0^2 x^2 F''(x)dx=40

, then

F'(2)+\int_0^2 F(x)dx

is equal to:

(A)11
(B)15
(C)9
(D)13

Q45·MathematicsSingle correctJEE Main 2025

If

\int_{-\pi/2}^{\pi/2}\frac{96x^2\cos^2 x}{1+e^x}dx=\pi(\alpha\pi^2+\beta)

,

\alpha, \beta\in Z

, then

(\alpha+\beta)^2

equals:

(A)144
(B)196
(C)100
(D)64

Q46·MathematicsSingle correctJEE Main 2025

Let f be a real valued continuous function defined on the positive real axis such that

g(x)=\int_0^x t\,f(t)dt

. If

g(x^3)=x^6+x^7

, then value of

\sum_{r=1}^{15} f(r^3)

is:

(A)320
(B)340
(C)270
(D)310

Q47·MathematicsSingle correctJEE Main 2025

In

I(m,n)=\int_0^1 x^{m-1}(1-x)^{n-1}dx

,

m,n>0

, then

I(9,14)+I(10,13)

is

(A) $I(9,1)$
(B) $I(19,27)$
(C) $I(1,13)$
(D) $I(9,13)$

Q48·MathematicsSingle correctJEE Main 2025

The value of

\displaystyle\int_{e^2}^{e^4}\frac{1}{x}\left(\frac{e^{\left((\log_e x)^2+1\right)^{-1}}}{e^{\left((\log_e x)^2+1\right)^{-1}}+e^{\left((6-\log_e x)^2+1\right)^{-1}}}\right)dx

is

(A) $\log_e 2$
(B) $2$
(C) $1$
(D) $e^2$

Q49·MathematicsSingle correctJEE Main 2025

If

I=\displaystyle\int_0^{\pi/2}\dfrac{\sin^2 x}{\sin^4 x+\cos^4 x}\,dx

, then

\displaystyle\int_0^{\pi/2}\dfrac{x\sin x\cos x}{\sin^4 x+\cos^4 x}\,dx

equals :

(A) $\dfrac{\pi^2}{16}$
(B) $\dfrac{\pi^2}{4}$
(C) $\dfrac{\pi^2}{8}$
(D) $\dfrac{\pi^2}{12}$

Q50·MathematicsSingle correctJEE Main 2025

Let

f(x)=7\tan^8x+7\tan^6x-3\tan^4x-3\tan^2x

,

I_1=\int_0^{\pi/4}f(x)\,dx

,

I_2=\int_0^{\pi/4}x\,f(x)\,dx

. Then

7I_1+12I_2

is:

(A) $2\pi$
(B) $\pi$
(C)1
(D)2

Q51·MathematicsIntegerJEE Advanced 2024

Let the function

f : [1, \infty) \to \mathbb{R}

be defined by

f(t) = \begin{cases} (-1)^{n+1} 2 & ; \text{ if } t = 2n - 1, n \in \mathbb{N} \\ \frac{(2n + 1 - t)}{2} f(2n - 1) + \frac{(t - (2n - 1))}{2} f(2n + 1) & ; \text{ if } 2n - 1 < t < 2n + 1, n \in \mathbb{N} \end{cases}

. Define

g(x) = \int_1^x f(t) \, dt

,

x \in (1, \infty)

. Let

\alpha

denote the number of solutions of the equation g(x) = 0 in the interval

(1, 8]

and

\beta = \lim_{x \to 1^+} \frac{g(x)}{x - 1}

. Then the value of

\alpha + \beta

is equal to ______

Q52·MathematicsNumericalJEE Advanced 2024

Let

f : \left[0, \frac{\pi}{2}\right] \to [0, 1]

be the function defined by

f(x) = \sin^2 x

and let

g : \left[0, \frac{\pi}{2}\right] \to [0, \infty)

be the function defined by

g = \sqrt{\frac{\pi x}{2} - x^2}

. The value of

\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x)g(x) \, dx

is ______

Q53·MathematicsNumericalJEE Advanced 2024

Let

f : \left[0, \frac{\pi}{2}\right] \to [0, 1]

be the function defined by

f(x) = \sin^2 x

and let

g : \left[0, \frac{\pi}{2}\right] \to [0, \infty)

be the function defined by

g = \sqrt{\frac{\pi x}{2} - x^2}

. The value of

2\int_0^{\frac{\pi}{2}} f(x)g(x) \, dx - \int_0^{\frac{\pi}{2}} g(x) \, dx

is ______

Q54·MathematicsNumericalJEE Main 2024

Let

\lim_{n \to \infty} \left( \frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \cdots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}} \right)

be

\frac{\pi}{k}

, using only the principal values of the inverse trigonometric functions. Then

k^2

is equal to ________.

Q55·MathematicsSingle correctJEE Main 2024

The integral

\displaystyle\int_{1/4}^{3/4}\cos\left(2\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right)dx

is equal to:

(A) $-\dfrac{1}{2}$
(B) $\dfrac{1}{4}$
(C) $\dfrac{1}{2}$
(D) $-\dfrac{1}{4}$

Q56·MathematicsSingle correctJEE Main 2024

The value of the integral

\displaystyle\int_{-1}^{2}\log_{e}\left(x+\sqrt{x^{2}+1}\right)dx

is:

(A) $\sqrt{5}-\sqrt{2}+\log_{e}\left(\dfrac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
(B) $\sqrt{2}-\sqrt{5}+\log_{e}\left(\dfrac{9+4\sqrt{5}}{1+\sqrt{2}}\right)$
(C) $\sqrt{5}-\sqrt{2}+\log_{e}\left(\dfrac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$
(D) $\sqrt{2}-\sqrt{5}+\log_{e}\left(\dfrac{7+4\sqrt{5}}{1+\sqrt{2}}\right)$

Q57·MathematicsSingle correctJEE Main 2024

Let

\displaystyle\int_{a}^{\log_e 4}\dfrac{dx}{\sqrt{e^x-1}}=\dfrac{\pi}{6}

. Then

e^a

and

e^{-a}

are the roots of the equation

(A) $2x^2-5x+2=0$
(B) $x^2-2x-8=0$
(C) $2x^2-5x-2=0$
(D) $x^2+2x-8=0$

Q58·MathematicsSingle correctJEE Main 2024

The value of

k\in\mathbb{N}

for which the integral

I_n=\displaystyle\int_0^1(1-x^k)^n\,dx

,

n\in\mathbb{N}

, satisfies

147\,I_{20}=148\,I_{21}

is:

(A) $10$
(B) $8$
(C) $14$
(D) $7$

Q59·MathematicsSingle correctJEE Main 2024

\displaystyle\int_{0}^{\pi/4}\frac{\cos^{2}x\,\sin^{2}x}{(\cos^{3}x+\sin^{3}x)^{2}}\,dx

is equal to

(A) $\tfrac{1}{12}$
(B) $\tfrac{1}{3}$
(C) $\tfrac{1}{6}$
(D) $\tfrac{1}{9}$

Q60·MathematicsNumericalJEE Main 2024

Let

r_{k}=\dfrac{\int_{0}^{1}(1-x^{7})^{k}\,dx}{\int_{0}^{1}(1-x^{7})^{k+1}\,dx}

,

k\in\mathbb{N}

. Then the value of

\displaystyle\sum_{k=1}^{10}\frac{1}{7(r_{k}-1)}

is equal to _______.

Q61·MathematicsNumericalJEE Main 2024

Let

[t]

denote the greatest integer less than or equal to

t

. Let

f:[0,\infty)\to R

be a function defined by

f(x)=\left[\frac{x}{2}+3\right]-\left[\sqrt{x}\right]

. Let

S

be the set of all points in the interval

[0,8]

at which

f

is not continuous. Then

\sum_{a\in S}a

is equal to ______.

Q62·MathematicsNumericalJEE Main 2024

Let

[t]

denote the largest integer less than or equal to

t

. If

\int_{0}^{3}\left([x^{2}]+\left[\frac{x^{2}}{2}\right]\right)dx=a+b\sqrt{2}-\sqrt{3}-\sqrt{5}+c\sqrt{6}-\sqrt{7}

, where

a,b,c\in Z

, then

a+b+c

is equal to ______.

Q63·MathematicsNumericalJEE Main 2024

If

f(t)=\displaystyle\int_0^{\pi}\dfrac{2x\,dx}{1-\cos^2 t\,\sin^2 x}

,

0<t<\pi

, then the value of

\displaystyle\int_0^{\pi/2}\dfrac{\pi^2\,dt}{f(t)}

equals __________.

Q64·MathematicsSingle correctJEE Main 2024

The integral

\displaystyle\int_{0}^{\pi/4} \dfrac{136\sin x}{3\sin x + 5\cos x}\,dx

is equal to:

(A) $3\pi - 50\log_e 2 + 20\log_e 5$
(B) $3\pi - 25\log_e 2 + 10\log_e 5$
(C) $3\pi - 10\log_e\left(2\sqrt{2}\right) + 10\log_e 5$
(D) $3\pi - 30\log_e 2 + 20\log_e 5$

Q65·MathematicsSingle correctJEE Main 2024

Let

\beta(m,n)=\displaystyle\int_0^1 x^{m-1}(1-x)^{n-1}\,dx

,

m,n>0

. If

\displaystyle\int_0^1 (1-x^{10})^{20}\,dx=a\,\beta(b,c)

, then

100(a+b+c)

equals:

(A)1021
(B)1120
(C)2012
(D)2120

Q66·MathematicsSingle correctJEE Main 2024

The value of

\displaystyle\int_{-\pi}^{\pi} \dfrac{2y(1+\sin y)}{1+\cos^2 y}\,dy

is:

(A) $\pi^2$
(B) $\dfrac{\pi^2}{2}$
(C) $\dfrac{\pi}{2}$
(D) $2\pi^2$

Q67·MathematicsSingle correctJEE Main 2024

Let

f(x)=\begin{cases}-2 & ,\ -2\le x\le0\\ x-2 & ,\ 0<x\le2\end{cases}

and

h(x)=f(|x|)+|f(x)|

. Then

\displaystyle\int_{-2}^{2}h(x)\,dx

is equal to:

(A) $2$
(B) $4$
(C) $1$
(D) $6$

Q68·MathematicsNumericalJEE Main 2024

If

\displaystyle\int_0^{\pi/4}\dfrac{\sin^2 x}{1+\sin x\cos x}\,dx=\dfrac1a\log_e\left(\dfrac a3\right)+\dfrac{\pi}{b\sqrt3}

, where

a,b\in\mathbb{N}

, then

a+b

is equal to ___

Q69·MathematicsSingle correctJEE Main 2024

If the value of the integral

\displaystyle\int_{-1}^{1}\dfrac{\cos\alpha x}{1+3^x}\,dx

is

\dfrac{2}{\pi}

, then a value of

\alpha

is

(A) $\tfrac{\pi}{6}$
(B) $\tfrac{\pi}{2}$
(C) $\tfrac{\pi}{3}$
(D) $\tfrac{\pi}{4}$

Q70·MathematicsSingle correctJEE Main 2024

The value of the integral

\displaystyle\int_0^{\pi/4}\dfrac{x\,dx}{\sin^4(2x)+\cos^4(2x)}

equals:

(A) $\dfrac{\sqrt2\,\pi^2}{8}$
(B) $\dfrac{\sqrt2\,\pi^2}{16}$
(C) $\dfrac{\sqrt2\,\pi^2}{32}$
(D) $\dfrac{\sqrt2\,\pi^2}{64}$

Q71·MathematicsNumericalJEE Main 2024

Let

f:(0,\infty)\to\mathbb R

and

F(x)=\displaystyle\int_0^x t f(t)\,dt

. If

F(x^2)=x^4+x^5

, then

\displaystyle\sum_{r=1}^{12} f(r^2)

is equal to __________.

Q72·MathematicsSingle correctJEE Main 2024

If

\displaystyle\int_0^{\pi/3}\cos^4 x\,dx=a\pi+b\sqrt 3

, where

a

and

b

are rational numbers, then

9a+8b

is equal to:

(A)2
(B)1
(C)3
(D) $\dfrac{3}{2}$

Q73·MathematicsNumericalJEE Main 2024

If

\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{8\sqrt2\cos x\,dx}{(1+e^{\sin x})(1+\sin^4 x)}=\alpha\pi+\beta\log_e(3+2\sqrt2)

, where

\alpha,\beta

are integers, then

\alpha^2+\beta^2

equals ___

Q74·MathematicsSingle correctJEE Main 2024

The value of

\displaystyle\int_0^1 (2x^3-3x^2-x+1)^{1/3}\,dx

is equal to:

(A)0
(B)1
(C)2
(D) $-1$

Q75·MathematicsNumericalJEE Main 2024

\left|\dfrac{120}{\pi^3}\displaystyle\int_0^{\pi}\dfrac{x^2\sin x\cos x}{\sin^4 x+\cos^4 x}\,dx\right|

is equal to ______.

Q76·MathematicsNumericalJEE Main 2024

Let

f:R\to R

be a function defined by

f(x)=\dfrac{4^x}{4^x+2}

and

M=\displaystyle\int_{f(a)}^{f(1-a)} x\sin^4\left(x(1-x)\right)dx

,

N=\displaystyle\int_{f(a)}^{f(1-a)}\sin^4\left(x(1-x)\right)dx

;

a\ne\dfrac{1}{2}

. If

\alpha M=\beta N

,

\alpha,\beta\in N

, then the least value of

\alpha^2+\beta^2

is equal to ______.

Q77·MathematicsNumericalJEE Main 2024

If the integral

525\displaystyle\int_0^{\pi/2}\sin 2x\cos^{11/2}x\left(1+\cos^{5/2}x\right)^{1/2}dx

is equal to

\left(n\sqrt{2}-64\right)

, then n is equal to ______.

Q78·MathematicsSingle correctJEE Main 2024

The value of

\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\dfrac{n^3}{(n^2+k^2)(n^2+3k^2)}

is:

(A) $\dfrac{(2\sqrt3-3)\pi}{24}$
(B) $\dfrac{13\pi}{8(4\sqrt3+3)}$
(C) $\dfrac{13(2\sqrt3-3)\pi}{24}$
(D) $\dfrac{\pi}{8(2\sqrt3+3)}$

Q79·MathematicsSingle correctJEE Main 2024

Let

y=f(x)

be a thrice differentiable function in

(-5,5)

. Let the tangents to the curve

y=f(x)

at

(1,f(1))

and

(3,f(3))

make angles

\dfrac{\pi}{6}

and

\dfrac{\pi}{4}

respectively with positive x-axis. If

27\displaystyle\int_1^3\left((f'(t))^2+1\right)f''(t)\,dt=\alpha+\beta\sqrt{3}

where

\alpha,\beta

are integers, then the value of

\alpha+\beta

equals:

(A) $-14$
(B) $26$
(C) $-16$
(D) $36$

Q80·MathematicsNumericalJEE Main 2024

The value

9\displaystyle\int_0^9\left[\sqrt{\dfrac{10x}{x+1}}\,\right]dx

, where

[t]

denotes the greatest integer less than or equal to

t

, is ___

Q81·MathematicsSingle correctJEE Main 2024

Let

f:R\to R

be a function defined

f(x)=\dfrac{x}{(1+x^4)^{1/4}}

and

g(x)=f(f(f(f(x))))

then

18\displaystyle\int_0^{\sqrt{2\sqrt{5}}}x^2\,g(x)\,dx

(A) $33$
(B) $36$
(C) $42$
(D) $39$

Q82·MathematicsSingle correctJEE Main 2024

Let

f:R\to R

be defined

f(x)=ae^{2x}+be^{x}+cx

. If

f(0)=-1

,

f'(\log_e 2)=21

and

\displaystyle\int_0^{\log_e 4}(f(x)-cx)\,dx=\dfrac{39}{2}

, then the value of

|a+b+c|

equals:

(A) $16$
(B) $10$
(C) $12$
(D) $8$

Q83·MathematicsSingle correctJEE Main 2024

If the value of the integral

\displaystyle\int_{-\pi/2}^{\pi/2}\left(\dfrac{x^2\cos x}{1+\pi^x}+\dfrac{1+\sin^2 x}{1+e^{\sin x^{2023}}}\right)dx=\dfrac{\pi}{4}(\pi+a)-2

, then the value of a is

(A) $3$
(B) $-\dfrac{3}{2}$
(C) $2$
(D) $\dfrac{3}{2}$

Q84·MathematicsNumericalJEE Main 2024

If

\displaystyle\int_{\pi/6}^{\pi/3}\sqrt{1-\sin 2x}\,dx=\alpha+\beta\sqrt2+\gamma\sqrt3

, where

\alpha,\beta

and

\gamma

are rational numbers, then

3\alpha+4\beta-\gamma

is equal to ___.

Q85·MathematicsSingle correctJEE Main 2024

For

0<a<1

, the value of the integral

\displaystyle\int_{0}^{\pi}\dfrac{dx}{1-2a\cos x+a^2}

is :

(A) $\dfrac{\pi^2}{\pi+a^2}$
(B) $\dfrac{\pi^2}{\pi-a^2}$
(C) $\dfrac{\pi}{1-a^2}$
(D) $\dfrac{\pi}{1+a^2}$

Q86·MathematicsSingle correctJEE Main 2024

If

(a,b)

be the orthocentre of the triangle whose vertices are

(1,2)

,

(2,3)

and

(3,1)

and

I_1=\displaystyle\int_a^b x\sin(4x-x^2)\,dx

,

I_2=\displaystyle\int_a^b \sin(4x-x^2)\,dx

, then

36\,\dfrac{I_1}{I_2}

is equal to:

(A)72
(B)88
(C)80
(D)66

Q87·MathematicsSingle correctJEE Main 2024

If

\displaystyle\int_0^1\dfrac{1}{\sqrt{3+x}+\sqrt{1+x}}\,dx=a+b\sqrt2+c\sqrt3

, where

a

,

b

,

c

are rational numbers, then

2a+3b-4c

is equal to:

(A)4
(B)10
(C)7
(D)8

Q88·MathematicsNumericalJEE Main 2024

Let

f(x)=\displaystyle\int_{0}^{x} g(t)\log_e\left(\dfrac{1-t}{1+t}\right)dt

, where

g

is a continuous odd function. If

\displaystyle\int_{-\pi/2}^{\pi/2}\left(f(x)+\dfrac{x^2\cos x}{1+e^x}\right)dx=\left(\dfrac{\pi}{\alpha}\right)^2-\alpha

, then

\alpha

is equal to __________.

Q89·MathematicsIntegerJEE Advanced 2023

For

x \in \mathbb{R}

, let

\tan^{-1}(x) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

. Then the minimum value of the function

f : \mathbb{R} \to \mathbb{R}

defined by

f(x) = \int_{0}^{x \tan^{-1} x} \frac{e^{(t - \cos t)}}{1 + t^{2023}}\,dt

is

Q90·MathematicsSingle correctJEE Main 2023

If

\int_0^1 \frac{1}{\left(5 + 2x - 2x^2\right)\left(1 + e^{(2-4x)}\right)}\,dx = \frac{1}{\alpha}\log_e\left(\frac{\alpha + 1}{\beta}\right)

,

\alpha, \beta > 0

, then

\alpha^4 - \beta^4

is equal to:

(A) $21$
(B) $0$
(C) $19$
(D) $-21$

Q91·MathematicsNumericalJEE Main 2023

Let for

x\in\mathbb{R},\,S_{0}(x)=x,\,S_{k}(x)=C_{k}+k\displaystyle\int_{0}^{x}S_{k-1}(t)dt

, where

C_{k}=1-\displaystyle\int_{0}^{1}S_{k-1}(x)dx,\,k=1,2,3,\dots

. Then

S_{2}(3)+6C_{3}

is equal to _____.

Q92·MathematicsNumericalJEE Main 2023

Let

f_n = \int_{0}^{\frac{\pi}{2}} \left(\sum\limits_{k=1}^{n} \sin^{k-1}x\right)\left(\sum\limits_{k=1}^{n} (2k-1)\sin^{k-1}x\right)\cos x\, dx

,

n \in \mathbb{N}

. Then

f_{21} - f_{20}

is equal to

Q93·MathematicsSingle correctJEE Main 2023

The value of

\dfrac{\displaystyle\int_{0}^{\frac{\pi}{4}} e^{-x} + \int_{0}^{\frac{\pi}{4}} e^{-x}\tan^{50}x\,dx}{\displaystyle\int_{0}^{\frac{\pi}{4}} e^{-x}\left(\tan^{49}x + \tan^{51}x\right)dx}

is

(A)50
(B)49
(C)51
(D)52

Q94·MathematicsSingle correctJEE Main 2023

\displaystyle\int_{0}^{2}\dfrac{6}{e^{3x}+6e^{2x}+11e^{x}+6}\,dx

is equal to:

(A) $\log_e\dfrac{512}{81}$
(B) $\log_e\dfrac{32}{27}$
(C) $\log_e\dfrac{256}{81}$
(D) $\log_e\dfrac{64}{27}$

Q95·MathematicsNumericalJEE Main 2023

If

\int_{-0.15}^{0.15}\left|100x^2-1\right|\,dx=\dfrac{k}{3000}

, then k is equal to _________.

Q96·MathematicsSingle correctJEE Main 2023

The value of the integral

\int_{-\log_e 2}^{\log_e 2} e^x \left( \log_e \left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx

is equal to

(A) $\log_e \left( \frac{2(2 + \sqrt{5})}{\sqrt{1 + \sqrt{5}}} \right) - \frac{\sqrt{5}}{2}$
(B) $\log_e \left( \frac{\sqrt{2}(3 - \sqrt{5})^2}{\sqrt{1 + \sqrt{5}}} \right) + \frac{\sqrt{5}}{2}$
(C) $\log_e \left( \frac{(2 + \sqrt{5})^2}{\sqrt{1 + \sqrt{5}}} \right) + \frac{\sqrt{5}}{2}$
(D) $\log_e \left( \frac{\sqrt{2}(2 + \sqrt{5})^2}{\sqrt{1 + \sqrt{5}}} \right) - \frac{\sqrt{5}}{2}$

Q97·MathematicsSingle correctJEE Main 2023

Let the function

f:[0,2]\to\mathbb{R}

be defined as

f(x)=\begin{cases} e^{\min\{x^2,\,x-[x]\}}, & x\in[0,1) \\ e^{[x-\log_e x]}, & x\in[1,2] \end{cases}

where

[t]

denotes the greatest integer less than or equal to

t

. Then the value of the integral

\int_0^2 x\,f(x)\,dx

is

(A) $2e-1$
(B) $2e+\dfrac{3e}{2}$
(C) $2e-\dfrac{1}{2}$
(D) $(e-1)\left(e^2+\dfrac{1}{2}\right)$

Q98·MathematicsNumericalJEE Main 2023

For

m, n > 0

, let

\alpha(m, n) = \int_0^2 t^m (1 + 3t)^n \, dt

. If

11\alpha(10, 6) + 18\alpha(11, 5) = p(14)^6

, then

p

is equal to _______ .

Q99·MathematicsSingle correctJEE Main 2023

If

f:\mathbb{R}\to\mathbb{R}

be a continuous function satisfying

\displaystyle\int_0^{\pi/2} f(\sin 2x)\sin x\,dx+\alpha\int_0^{\pi/4} f(\cos 2x)\cos x\,dx=0

, then

\alpha

is equal to

(A) $-\sqrt{3}$
(B) $\sqrt{2}$
(C) $\sqrt{3}$
(D) $-\sqrt{2}$

Q100·MathematicsSingle correctJEE Main 2023

Let

f

be a continuous function satisfying

\int_0^{t^2}\left(f(x)+x^2\right)dx=\frac{4}{3}t^3,\ \forall\ t>0

. Then

f\left(\frac{\pi^2}{4}\right)

is equal to:

(A) $\pi\left(1-\frac{\pi^3}{16}\right)$
(B) $-\pi^2\left(1+\frac{\pi^2}{16}\right)$
(C) $-\pi\left(1+\frac{\pi^2}{16}\right)$
(D) $\pi^2\left(1-\frac{\pi^2}{16}\right)$

Q101·MathematicsSingle correctJEE Main 2023

Let

f

be a differentiable function such that

x^2 f(x)-x=4\int_0^x t\,f(t)\,dt

,

f(1)=\dfrac23

. Then

18\,f(3)

is equal to

(A) $160$
(B) $210$
(C) $180$
(D) $150$

Q102·MathematicsNumericalJEE Main 2023

Let

[t]

denotes the greatest integer

\le t

. Then

\frac{2}{\pi}\int_{\pi/6}^{5\pi/6}\left(8[\operatorname{cosec} x]-5[\cot x]\right)dx

is equal to

Q103·MathematicsNumericalJEE Main 2023

Let

[t]

denote the greatest integer function. If

\displaystyle\int_0^{2.4}\left[x^{2}\right]dx=\alpha+\beta\sqrt{2}+\gamma\sqrt{3}+\delta\sqrt{5}

, then

\alpha+\beta+\gamma+\delta

is equal to

Q104·MathematicsSingle correctJEE Main 2023

Let

5f(x)+4f\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+3,\,x>0

. Then

18\displaystyle\int_{1}^{2}f(x)\,dx

is equal to:

(A) $10\log_e 2-6$
(B) $10\log_e 2+6$
(C) $6\log_e 2+3$
(D) $6\log_e 2-3$

Q105·MathematicsNumericalJEE Main 2023

Let

f(x)=\frac{x}{\left(1+x^n\right)^{\frac{1}{n}}}

,

x\in \mathbb{R}-\{-1\}

,

n\in \mathbb{N}

,

n>2

. If

f^n(x)=(\underbrace{f\circ f\circ \cdots \circ f}_{n\text{ times}})(x)

, then

\lim_{n\to\infty}\int x^{n-2}\left(f^n(x)\right)dx

is equal to

Q106·MathematicsSingle correctJEE Main 2023

Let

f(x)

be a function satisfying

f(x)+f(\pi-x)=\pi^2

,

\forall x\in \mathbb{R}

. Then

\int_0^\pi f(x)\sin x\,dx

is equal to

(A) $\frac{\pi^2}{4}$
(B) $\frac{\pi^2}{2}$
(C) $2\pi^2$
(D) $\pi^2$

Q107·MathematicsSingle correctJEE Main 2023

\lim_{n\to\infty}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\cdots+\frac{1}{2n}\right]

is equal to

(A) $\log_e 2$
(B) $\log_e\!\left(\frac{2}{3}\right)$
(C) $\log_e\!\left(\frac{3}{2}\right)$
(D) $0$

Q108·MathematicsNumericalJEE Main 2023

If

\int_0^1 (x^{21}+x^{14}+x^7)(2x^{14}+3x^7+6)^{1/7}\,dx=\frac{1}{l}(11)^{m/n}

where

l,m,n\in\mathbb{N}

,

m

and

n

are coprime, then

l+m+n

is equal to _______ .

Q109·MathematicsNumericalJEE Main 2023

If

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{5^{\cos x}\left(1+\cos x\cos 3x+\cos^2 x+\cos^3 x\cos 3x\right)}{1+5^{\cos x}}dx=\dfrac{k\pi}{16}

, then k is equal to

Q110·MathematicsSingle correctJEE Main 2023

The value of the integral

\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\dfrac{x+\frac{\pi}{4}}{2-\cos 2x}dx

is:

(A) $\dfrac{\pi^2}{12\sqrt3}$
(B) $\dfrac{\pi^2}{6\sqrt3}$
(C) $\dfrac{\pi^2}{6}$
(D) $\dfrac{\pi^2}{3\sqrt3}$

Q111·MathematicsSingle correctJEE Main 2023

If

\phi(x)=\dfrac{1}{\sqrt{x}}\displaystyle\int_{\frac{\pi}{4}}^{x}\left(4\sqrt2\sin t-3\phi'(t)\right)dt,\ x>0

, then

\phi'\left(\dfrac{\pi}{4}\right)

is equal to:

(A) $\dfrac{8}{6+\sqrt\pi}$
(B) $\dfrac{4}{6+\sqrt\pi}$
(C) $\dfrac{8}{\sqrt\pi}$
(D) $\dfrac{4}{6-\sqrt\pi}$

Q112·MathematicsSingle correctJEE Main 2023

The value of

\int_{\pi/3}^{\pi/2}\frac{(2+3\sin x)}{\sin x(1+\cos x)}\,dx

is equal to

(A) $\frac{10}{3}-\sqrt{3}-\log_{e}\sqrt{3}$
(B) $\frac{7}{2}-\sqrt{3}-\log_{e}\sqrt{3}$
(C) $-2+3\sqrt{3}+\log_{e}\sqrt{3}$
(D) $\frac{10}{3}-\sqrt{3}+\log_{e}\sqrt{3}$

Q113·MathematicsSingle correctJEE Main 2023

Let

\alpha>0

. If

\displaystyle\int_{0}^{\alpha}\dfrac{x}{\sqrt{x+\alpha}-\sqrt{x}}dx=\dfrac{16+20\sqrt2}{15}

, then

\alpha

is equal to:

(A) $4$
(B) $2\sqrt2$
(C) $\sqrt2$
(D) $2$

Q114·MathematicsSingle correctJEE Main 2023

If

[t]

denotes the greatest integer

\leq t

, then the value of

\dfrac{3(e-1)}{e}\int_1^2 x^2 e^{[x^3]+[x^4]}\,dx

is:

(A) $e^8-1$
(B) $e^7-1$
(C) $e^8-e$
(D) $e^9-e$

Q115·MathematicsSingle correctJEE Main 2023

\displaystyle\lim_{n \to \infty} \dfrac{3}{n}\left\{4 + \left(2 + \dfrac{1}{n}\right)^2 + \left(2 + \dfrac{2}{n}\right)^2 + \cdots + \left(3 - \dfrac{1}{n}\right)^2\right\}

is equal to:

(A) $12$
(B) $\dfrac{19}{3}$
(C) $0$
(D) $19$

Q116·MathematicsSingle correctJEE Main 2023

The value of the integral

\displaystyle\int_{1/2}^{2}\dfrac{\tan^{-1}x}{x}dx

is equal to:

(A) $\dfrac{\pi}{2}\log_e 2$
(B) $\pi\log_e 2$
(C) $\dfrac12\log_e 2$
(D) $\dfrac{\pi}{4}\log_e 2$

Q117·MathematicsSingle correctJEE Main 2023

Let

[x]

denote the greatest integer

\leq x

. Consider the function

f(x) = \max\{x^2, 1 + [x]\}

. Then the value of the integral

\int_0^2 f(x)\,dx

is

(A) $\dfrac{5 + 4\sqrt{2}}{3}$
(B) $\dfrac{4 + 5\sqrt{2}}{3}$
(C) $\dfrac{1 + 5\sqrt{2}}{3}$
(D) $\dfrac{8 + 4\sqrt{2}}{3}$

Q118·MathematicsSingle correctJEE Main 2023

Let

f(x) = x + \dfrac{a}{\pi^2 - 4}\sin x + \dfrac{b}{\pi^2 - 4}\cos x

,

x \in \mathbb{R}

be a function which satisfies

f(x) = x + \int_0^{\pi/2} \sin(x + y) f(y)\,dy

. Then

(a + b)

is equal to

(A) $-2\pi(\pi - 2)$
(B) $-2\pi(\pi + 2)$
(C) $-\pi(\pi - 2)$
(D) $-\pi(\pi + 2)$

Q119·MathematicsSingle correctJEE Main 2023

The value of the integral

\displaystyle\int_{1}^{2}\left(\dfrac{t^{4}+1}{t^{6}+1}\right)dt

is:

(A) $\tan^{-1}2-\dfrac13\tan^{-1}8+\dfrac{\pi}{4}$
(B) $\tan^{-1}\dfrac12+\dfrac13\tan^{-1}8-\dfrac{\pi}{3}$
(C) $\tan^{-1}\dfrac12-\dfrac13\tan^{-1}8+\dfrac{\pi}{3}$
(D) $\tan^{-1}2+\dfrac13\tan^{-1}8-\dfrac{\pi}{3}$

Q120·MathematicsSingle correctJEE Main 2023

The minimum value of the function

f(x)=\displaystyle\int_0^2 e^{|x-t|}\,dt

is:

(A) $e(e-1)$
(B) $2(e-1)$
(C) $2$
(D) $2e-1$

Q121·MathematicsNumericalJEE Main 2023

If

\displaystyle\int_{1/3}^{3}|\log_{e}x|\,dx=\dfrac{m}{n}\log_{e}\left(\dfrac{n^{2}}{e}\right)

, where

m

and

n

are coprime natural numbers, then

m^{2}+n^{2}-5

is equal to

Q122·MathematicsSingle correctJEE Main 2023

The integral

16\displaystyle\int_{1}^{2} \dfrac{dx}{x^3 (x^2 + 2)^2}

is equal to

(A) $\dfrac{11}{12} - \log_e 4$
(B) $\dfrac{11}{6} - \log_e 4$
(C) $\dfrac{11}{6} + \log_e 4$
(D) $\dfrac{11}{12} + \log_e 4$

Q123·MathematicsNumericalJEE Main 2023

The value of

12\displaystyle\int_0^3 |x^2-3x+2|\,dx

is _______ .

Q124·MathematicsNumericalJEE Main 2023

The value of

\dfrac{8}{\pi}\displaystyle\int_0^{\pi/2}\dfrac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}}\,dx

is _______ .

Q125·MathematicsSingle correctJEE Main 2023

\displaystyle\int_{\frac{3\sqrt2}{4}}^{\frac{3\sqrt3}{4}}\dfrac{48}{\sqrt{9-4x^2}}\,dx

is equal to

(A) $2\pi$
(B) $\dfrac{\pi}{6}$
(C) $\dfrac{\pi}{3}$
(D) $\dfrac{\pi}{2}$

Q126·MathematicsIntegerJEE Advanced 2022

The greatest integer less than or equal to

\int_{1}^{2} \log_{2}\left(x^{3} + 1\right)dx + \int_{1}^{\log_{2}9} \left(2^{x} - 1\right)^{1/3} dx

is ______.

Q127·MathematicsSingle correctJEE Advanced 2022

For positive integer n, define f(n) = n +

\frac{16+5n-3n^{2}}{4n+3n^{2}}

+

\frac{32+n-3n^{2}}{8n+3n^{2}}

+

\frac{48-3n-3n^{2}}{12n+3n^{2}}

+ … +

\frac{25n-7n^{2}}{7n^{2}}

. Then, the value of

\lim_{n \to \infty}

f(n) is equal to

(A)3 + $\frac{4}{3}$ log $_{e}$ 7
(B)4 − $\frac{3}{4}$ log $_{e}$ $\left(\frac{7}{3}\right)$
(C)4 − $\frac{4}{3}$ log $_{e}$ $\left(\frac{7}{3}\right)$
(D)3 + $\frac{3}{4}$ log $_{e}$ 7 .

Q128·MathematicsMultiple correctJEE Advanced 2022

Consider the equation

\int_{1}^{e} \frac{\left(\log_{e} x\right)^{\frac{1}{2}}}{x\left(a - \left(\log_{e} x\right)^{\frac{3}{2}}\right)^{2}} \, dx = 1

,

a \in \left(-\infty, 0\right) \cup \left(1, \infty\right)

. Which of the following statements is/are TRUE?

(A)No $a$ satisfies the above equation
(B)An integer $a$ satisfies the above equation
(C)An irrational number $a$ satisfies the above equation
(D)More than one $a$ satisfy the above equation

Q129·MathematicsSingle correctJEE Main 2022

The integral

\int_{0}^{\frac{\pi}{2}} \frac{1}{3 + 2\sin x + \cos x}\,dx

is equal to:

(A) $\tan^{-1}(2)$
(B) $\tan^{-1}(2) - \frac{\pi}{4}$
(C) $\frac{1}{2}\tan^{-1}(2) - \frac{\pi}{8}$
(D) $\frac{1}{2}$

Q130·MathematicsSingle correctJEE Main 2022

If

[t]

denotes the greatest integer

\leq t

, then the value of

\int_{0}^{1} \left[ 2x - |3x^2 - 5x + 2| + 1 \right] dx

is:

(A) $\frac{\sqrt{37} + \sqrt{13} - 4}{6}$
(B) $\frac{\sqrt{37} - \sqrt{13} - 4}{6}$
(C) $\frac{-\sqrt{37} - \sqrt{13} + 4}{6}$
(D) $\frac{-\sqrt{37} + \sqrt{13} + 4}{6}$

Q131·MathematicsSingle correctJEE Main 2022

If

f(\alpha) = \int_{1}^{\alpha} \frac{\log_{10} t}{1 + t}\,dt, \alpha > 0

, then

f(e^{3}) + f(e^{-3})

is equal to :

(A)9
(B) $\frac{9}{2}$
(C) $\frac{9}{\log_{e}(10)}$
(D) $\frac{9}{2\log_{e}(10)}$

Q132·MathematicsSingle correctJEE Main 2022

Let

I_{n}(x)=\int_{0}^{x}\frac{1}{\left(t^{2}+5\right)^{n}}dt

,

n=1,2,3,\ldots.

Then

(A) $50I_{6}-9I_{5}=xI_{5}'$
(B) $50I_{6}-11I_{5}=xI_{5}'$
(C) $50I_{6}-9I_{5}=I_{5}'$
(D) $50I_{6}-11I_{5}=I_{5}'$

Q133·MathematicsSingle correctJEE Main 2022

The minimum value of the twice differentiable function

f(x) = \int_{0}^{x} e^{x-t} f'(t)\,dt - (x^{2} - x + 1)e^{x}

,

x \in R

, is :

(A) $-\frac{2}{\sqrt{e}}$
(B) $-2\sqrt{e}$
(C) $-\sqrt{e}$
(D) $\frac{2}{\sqrt{e}}$

Q134·MathematicsNumericalJEE Main 2022

The value of the integral

\int_{0}^{\pi/2}60\frac{\sin(6x)}{\sin x}dx

is equal to

Q135·MathematicsNumericalJEE Main 2022

If

\int_{0}^{\sqrt{3}} \frac{15x^{3}}{\sqrt{1 + x^{2} + \sqrt{\left(1 + x^{2}\right)^{3}}}}\,dx = \alpha\sqrt{2} + \beta\sqrt{3}

, where

\alpha

,

\beta

are integers, then

\alpha + \beta

is equal to ______.

Q136·MathematicsSingle correctJEE Main 2022

I = \int_{\pi/4}^{\pi/3}\left(\frac{8\sin x - \sin 2x}{x}\right)dx

. Then

(A) $\frac{\pi}{2} < I < \frac{3\pi}{4}$
(B) $\frac{\pi}{5} < I < \frac{5\pi}{12}$
(C) $\frac{5\pi}{12} < I < \frac{\sqrt{2}}{3}\pi$
(D) $\frac{3\pi}{4} < I < \pi$

Q137·MathematicsNumericalJEE Main 2022

Let

f(x) = \min\{[x-1], [x-2], ..., [x-10]\}

where

[t]

denotes the greatest integer

\le t

. Then

\int_{0}^{10} f(x)dx + \int_{0}^{10} (f(x))^{2}dx + \int_{0}^{10} |f(x)|dx

is equal to ___.

Q138·MathematicsNumericalJEE Main 2022

Let f be a differentiable function satisfying

f(x) = \frac{2}{\sqrt{3}}\int_{0}^{\sqrt{3}} f\left(\frac{\lambda^{2}x}{3}\right)d\lambda

,

x > 0

and

f(1) = \sqrt{3}

. If

y = f(x)

passes through the point

(\alpha, 6)

, then

\alpha

is equal to _______.

Q139·MathematicsSingle correctJEE Main 2022

Let

f : \mathbb{R} \to \mathbb{R}

be a function defined as

f(x) = a\sin\left(\frac{\pi[x]}{2}\right) + [2-x]

,

a \in \mathbb{R}

, where [t] is the greatest integer less than or equal to

t

. If

\lim_{x \to -1} f(x)

exists, then the value of

\int_{0}^{4} f(x)\,dx

is equal to :

(A)-1
(B)-2
(C)1
(D)2

Q140·MathematicsSingle correctJEE Main 2022

\int_{0}^{2}\left(\left|2x^{2}-3x\right|+\left[x-\frac{1}{2}\right]\right)dx

, where

[t]

is the greatest integer function, is equal to:

(A) $\frac{7}{6}$
(B) $\frac{19}{12}$
(C) $\frac{31}{12}$
(D) $\frac{3}{2}$

Q141·MathematicsSingle correctJEE Main 2022

\int\limits_{0}^{20\pi}\left(\left|\sin x\right|+\left|\cos x\right|\right)^{2}dx

is equal to :-

(A) $10\left(\pi+4\right)$
(B) $10\left(\pi+2\right)$
(C) $20\left(\pi-2\right)$
(D) $20\left(\pi+2\right)$

Q142·MathematicsNumericalJEE Main 2022

If

n\left(2n+1\right)\int_{0}^{1}\left(1-x^{n}\right)^{2n}\,dx = 1177\int_{0}^{1}\left(1-x^{n}\right)^{2n+1}dx

, then n ∈ N is equal to _______

Q143·MathematicsSingle correctJEE Main 2022

If a =

\lim\limits_{n \to \infty}\sum\limits_{k=1}^{n}\dfrac{2n}{n^{2}+k^{2}}

and f(x) =

\sqrt{\dfrac{1-\cos x}{1+\cos x}}

, x ∈ (0,1), then :

(A) $2\sqrt{2}\,\mathrm{f}\left(\dfrac{a}{2}\right) = \mathrm{f}\,'\left(\dfrac{a}{2}\right)$
(B) $\mathrm{f}\left(\dfrac{a}{2}\right)\mathrm{f}\,'\left(\dfrac{a}{2}\right) = \sqrt{2}$
(C) $\sqrt{2}\,\mathrm{f}\left(\dfrac{a}{2}\right) = \mathrm{f}\,'\left(\dfrac{a}{2}\right)$
(D) $\mathrm{f}\left(\dfrac{a}{2}\right) = \sqrt{2}\,\mathrm{f}\,'\left(\dfrac{a}{2}\right)$

Q144·MathematicsSingle correctJEE Main 2022

For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real valued function defined on the interval

[-10, 10]

by

f(x) = \begin{cases} x - [x], & \text{if}(x) \text{ is odd} \\ 1 + [x] - x & \text{if}(x) \text{ is even} \end{cases}

Then the value of

\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x\, dx

is :

(A)4
(B)2
(C)1
(D)0

Q145·MathematicsNumericalJEE Main 2022

If

\lim_{n\to\infty} \frac{(n+1)^{k-1}}{n^{k+1}} [(nk + 1) + (nk + 2) + \ldots + (nk + n)] = 33. \lim_{n\to\infty} \frac{1}{n^{k+1}} \cdot [1^{k} + 2^{k} + 3^{k} + \ldots + n^{k}]

, then the integral value of k is equal to ________.

Q146·MathematicsSingle correctJEE Main 2022

\int_{0}^{5} \cos\left(\pi\left(x - \left[\frac{x}{2}\right]\right)\right) dx

, Where [t] denotes greatest integer less than or equal to t, is equal to :

(A)−3
(B)−2
(C)2
(D)0

Q147·MathematicsSingle correctJEE Main 2022

Let

f

be a real valued continuous function on

[0,1]

and

f(x) = x + \int_{0}^{1}(x - t)f(t)dt

. Then which of the following points (x,y) lies on the curve

y = f(x)

?

(A) $(2, 4)$
(B) $(1, 2)$
(C) $(4, 17)$
(D) $(6, 8)$

Q148·MathematicsSingle correctJEE Main 2022

Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral

\int_{0}^{1}[-8x^{2}+6x-1]\,dx

is equal to

(A) $-1$
(B) $-\frac{5}{4}$
(C) $\frac{\sqrt{17}-13}{8}$
(D) $\frac{\sqrt{17}-16}{8}$

Q149·MathematicsSingle correctJEE Main 2022

Let

f : \mathbb{R} \to \mathbb{R}

be a differentiable function such that

f\left(\frac{\pi}{4}\right) = \sqrt{2}, f\left(\frac{\pi}{2}\right) = 0

and

f'\left(\frac{\pi}{2}\right) = 1

and let

g(x) = \int_{x}^{\pi/4} \left(f'(t)\sec t + \tan t \sec t\, f(t)\right) dt

for

x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right)

. Then

\lim_{x \to \left(\frac{\pi}{2}\right)^{-}} g(x)

is equal to

(A)2
(B)3
(C)4
(D) $-3$

Q150·MathematicsSingle correctJEE Main 2022

Let

f : \mathbb{R} \to \mathbb{R}

be continuous function satisfying

f(x) + f(x + k) = n

, for all

x \in \mathbb{R}

where

k > 0

and n is a positive integer. If

I_1 = \int_{0}^{4nk} f(x)dx

and

I_2 = \int_{-k}^{3k} f(x)dx

, then

(A) $I_1 + 2I_2 = 4nk$
(B) $I_1 + 2I_2 = 2nk$
(C) $I_1 + nI_2 = 4n^{2}k$
(D) $I_1 + nI_2 = 6n^{2}k$

Q151·MathematicsSingle correctJEE Main 2022

The value of the integral

\int_{-2}^{2} \frac{\left|x^{3} + x\right|}{\left(e^{x|x|} + 1\right)}dx

is equal to :

(A) $5e^{2}$
(B) $3e^{-2}$
(C)4
(D)6

Q152·MathematicsSingle correctJEE Main 2022

Let f be a differentiable function in

\left(0,\dfrac{\pi}{2}\right)

. If

\int_{\cos x}^{1}t^{2}f(t)dt=\sin^{3}x+\cos x

then

\dfrac{1}{\sqrt{3}}f'\left(\dfrac{1}{\sqrt{3}}\right)

is equal to :

(A) $6-9\sqrt{2}$
(B) $6-\dfrac{9}{\sqrt{2}}$
(C) $\dfrac{9}{2}-6\sqrt{2}$
(D) $\dfrac{9}{\sqrt{2}}-6$

Q153·MathematicsSingle correctJEE Main 2022

The integral

\int_{0}^{1}\dfrac{1}{7^{\left[\frac{1}{x}\right]}}dx

, where [.] denotes the greatest integer function is equal to

(A) $1+6\log_{e}\left(\dfrac{6}{7}\right)$
(B) $1-6\log_{e}\left(\dfrac{6}{7}\right)$
(C) $\log_{e}\left(\dfrac{7}{6}\right)$
(D) $1-7\log_{e}\left(\dfrac{6}{7}\right)$

Q154·MathematicsNumericalJEE Main 2022

The integral

\frac{24}{\pi}\int_{0}^{\sqrt{2}}\frac{\left(2-x^{2}\right)dx}{\left(2+x^{2}\right)\sqrt{4+x^{4}}}

is equal to ____.

Q155·MathematicsNumericalJEE Main 2022

Let

f(x) = \max\{|x + 1|, |x + 2|, ..., |x + 5|\}

. Then

\int_{-6}^{0} f(x)\,dx

is equal to ______________.

Q156·MathematicsNumericalJEE Main 2022

The value of the integral

\frac{48}{\pi^{4}}\int_{0}^{\pi}\left(\frac{3\pi x^{2}}{2} - x^{3}\right)\frac{\sin x}{1 + \cos^{2} x}\,dx

is equal to ________.

Q157·MathematicsSingle correctJEE Main 2022

The value of

\int_0^{\pi} \frac{e^{\cos x}\sin x}{(1+\cos^2 x)(e^{\cos x} + e^{-\cos x})}dx

is equal to

(A) $\frac{\pi^2}{4}$
(B) $\frac{\pi^2}{2}$
(C) $\frac{\pi}{4}$
(D) $\frac{\pi}{2}$

Q158·MathematicsNumericalJEE Main 2022

The value of b > 3 for which

12\int_{3}^{b}\frac{1}{\left(x^{2} - 1\right)\left(x^{2} - 4\right)}dx = \log_{e}\left(\frac{49}{40}\right)

, is equal to

Q159·MathematicsSingle correctJEE Main 2022

If

b_n = \int_{0}^{\frac{\pi}{2}} \frac{\cos^2 nx}{\sin x} dx

,

n \in N

, then

(A) $b_3 - b_2$ , $b_4 - b_3$ , $b_5 - b_4$ are in an A.P. with common difference $-2$
(B) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference 2
(C) $b_3 - b_2$ , $b_4 - b_3$ , $b_5 - b_4$ are in a G.P.
(D) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $-2$

Q160·MathematicsSingle correctJEE Main 2022

\lim_{n\to\infty}\left(\dfrac{n^{2}}{\left(n^{2}+1\right)\left(n+1\right)} + \dfrac{n^{2}}{\left(n^{2}+4\right)\left(n+2\right)} + \dfrac{n^{2}}{\left(n^{2}+9\right)\left(n+3\right)} + ... + \dfrac{n^{2}}{\left(n^{2}+n^{2}\right)\left(n+n\right)}\right)

is equal to

(A) $\dfrac{\pi}{8}+\dfrac{1}{4}\log_e 2$
(B) $\dfrac{\pi}{4}+\dfrac{1}{8}\log_e 2$
(C) $\dfrac{\pi}{4}-\dfrac{1}{8}\log_e 2$
(D) $\dfrac{\pi}{8}+\log_e \sqrt{2}$

Q161·MathematicsNumericalJEE Main 2022

Let

f(\theta)=\sin\theta+\int_{-\pi/2}^{\pi/2}(\sin\theta+t\cos\theta)f(t)dt

. Then the value of

\left|\int_{0}^{\pi/2} f(\theta)d\theta\right|

is ______ .

Q162·MathematicsSingle correctJEE Main 2022

The value of the integral

\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{dx}{\left(1+e^{x}\right)\left(\sin^{6} x+\cos^{6} x\right)}

is equal to

(A) $2\pi$
(B) $0$
(C) $\pi$
(D) $\dfrac{\pi}{2}$

Q163·MathematicsNumericalJEE Main 2022

Let

\underset{0\le x\le 2}{\mathrm{Max}}\left\{\dfrac{9-x^{2}}{5-x}\right\}=\alpha

and

\underset{0\le x\le 2}{\mathrm{Min}}\left\{\dfrac{9-x^{2}}{5-x}\right\}=\beta

If

\displaystyle\int_{\beta-\frac{8}{3}}^{2\alpha-1}\mathrm{Max}\left\{\dfrac{9-x^{2}}{5-x},x\right\}dx=\alpha_{1}+\alpha_{2}\log_{e}\left(\dfrac{8}{15}\right)

then

\alpha_{1}+\alpha_{2}

is equal to _______

Q164·MathematicsNumericalJEE Advanced 2021

Let

g_i : \left[ \frac{\pi}{8}, \frac{3\pi}{8} \right] \to \mathbb{R}

,

i = 1, 2

, and

f : \left[ \frac{\pi}{8}, \frac{3\pi}{8} \right] \to \mathbb{R}

be functions such that

g_1(x) = 1

,

g_2(x) = |4x - \pi|

and

f(x) = \sin^2 x

, for all

x \in \left[ \frac{\pi}{8}, \frac{3\pi}{8} \right]

Define

S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx

,

i = 1, 2

The value of

\frac{16S_1}{\pi}

is _____.

Q165·MathematicsMultiple correctJEE Advanced 2021

Let

f : \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \to \mathbb{R}

be a continuous function such that

f(0) = 1

and

\int_0^{\frac{\pi}{3}} f(t)\, dt = 0

Then which of the following statements is (are) TRUE?

(A)The equation $f(x) - 3\cos 3x = 0$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
(B)The equation $f(x) - 3\sin 3x = -\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
(C) $\lim_{x \to 0} \frac{x\int_0^x f(t)dt}{1 - e^{x^2}} = -1$
(D) $\lim_{x \to 0} \frac{\sin x \int_0^x f(t)dt}{x^2} = -1$

Q166·MathematicsNumericalJEE Advanced 2021

Let

g_i : \left[ \frac{\pi}{8}, \frac{3\pi}{8} \right] \to \mathbb{R}

,

i = 1, 2

, and

f : \left[ \frac{\pi}{8}, \frac{3\pi}{8} \right] \to \mathbb{R}

be functions such that

g_1(x) = 1

,

g_2(x) = |4x - \pi|

and

f(x) = \sin^2 x

, for all

x \in \left[ \frac{\pi}{8}, \frac{3\pi}{8} \right]

Define

S_i = \int_{\pi/8}^{3\pi/8} f(x) \cdot g_i(x) \, dx

,

i = 1, 2

The value of

\frac{48S_2}{\pi^2}

is _____.

Q167·MathematicsSingle correctJEE Advanced 2021

Let

\psi_1 : [0, \infty) \to \mathbb{R}

,

\psi_2 : [0, \infty) \to \mathbb{R}

,

f : [0, \infty) \to \mathbb{R}

and

g : [0, \infty) \to \mathbb{R}

be functions such that

f(0) = g(0) = 0,

\psi_1(x) = e^{-x} + x, \quad x \ge 0,

\psi_2(x) = x^2 - 2x - 2e^{-x} + 2, \quad x \ge 0,

f(x) = \int_{-x}^{x} \left( |t| - t^2 \right) e^{-t^2} \, dt, \quad x > 0

and

g(x) = \int_{0}^{x^2} \sqrt{t} \; e^{-t} \, dt, \quad x > 0

Which of the following statements is TRUE ?

(A) $\psi_1(x) \le 1$ , for all x > 0
(B) $\psi_2(x) \le 0$ , for all x > 0
(C) $f(x) \ge 1 - e^{-x^2} - \frac{2}{3}x^3 + \frac{2}{5}x^5$ , for all $x \in \left(0, \frac{1}{2}\right)$
(D) $g(x) \le \frac{2}{3}x^3 - \frac{2}{5}x^5 + \frac{1}{7}x^7$ , for all $x \in \left(0, \frac{1}{2}\right)$

Q168·MathematicsIntegerJEE Advanced 2021

For any real number x, let [x] denote the largest integer less than or equal to x. If

I = \int_0^{10} \left[\sqrt{\frac{10x}{x+1}}\right] dx,

then the value of 9I is _____.

Q169·MathematicsSingle correctJEE Main 2021

The function f(x), that satisfies the condition

f(x) = x + \int_{0}^{\pi/2} \sin x \cdot \cos y\, f(y)dy

, is :

(A) $x + \frac{2}{3}(\pi - 2)\sin x$
(B)x + (π + 2) sinx
(C) $x + \frac{\pi}{2}\sin x$
(D)x + (π − 2) sinx

Q170·MathematicsNumericalJEE Main 2021

If

x\,\phi(x) = \int_{5}^{x} \left( 3t^{2} - 2\phi'(t) \right) dt

,

x > -2

, and

\phi(0) = 4

, then

\phi(2)

is ______.

Q171·MathematicsSingle correctJEE Main 2021

Let f be a non-negative function in

[0, 1]

and twice differentiable in

(0, 1)

. If

\int_{0}^{x} \sqrt{1 - \left(f'(t)\right)^{2}}\, dt = \int_{0}^{x} f(t)\, dt

,

0 \le x \le 1

and

f(0) = 0

, then

\lim_{x \to 0} \frac{1}{x^{2}} \int_{0}^{x} f(t)\, dt

:

(A)equals 0
(B)equals 1
(C)does not exist
(D)equals $\frac{1}{2}$

Q172·MathematicsSingle correctJEE Main 2021

If

[x]

is the greatest integer

\leq x

, then

\pi^2 \int_{0}^{2} \left(\sin \frac{\pi x}{2}\right)(x - [x])^{[x]} \, dx

is equal to :

(A) $2(\pi - 1)$
(B) $4(\pi - 1)$
(C) $4(\pi + 1)$
(D) $2(\pi + 1)$

Q173·MathematicsNumericalJEE Main 2021

Let [t] denote the greatest integer

\le t

. Then the value of

8 \cdot \int_{-\frac{1}{2}}^{1} \left( [2x] + |x| \right) dx

is ______.

Q174·MathematicsSingle correctJEE Main 2021

\int_{6}^{16} \frac{\log_e x^2}{\log_e x^2 + \log_e\left(x^2 - 44x + 484\right)} dx

is equal to:

(A) $6$
(B) $8$
(C) $5$
(D) $10$

Q175·MathematicsSingle correctJEE Main 2021

The value of the integral

\int_{0}^{1} \frac{\sqrt{x}\,dx}{(1+x)(1+3x)(3+x)}

is:

(A) $\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{2}\right)$
(B) $\frac{\pi}{4}\left(1 - \frac{\sqrt{3}}{6}\right)$
(C) $\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{6}\right)$
(D) $\frac{\pi}{4}\left(1 - \frac{\sqrt{3}}{2}\right)$

Q176·MathematicsSingle correctJEE Main 2021

The value of

\int_{-1/\sqrt{2}}^{1/\sqrt{2}} \left(\left(\frac{x+1}{x-1}\right)^2 + \left(\frac{x-1}{x+1}\right)^2 - 2\right)^{1/2} dx

is:

(A) $\log_e 4$
(B) $\log_e 16$
(C) $2\log_e 16$
(D) $4\log_e (3 + 2\sqrt{2})$

Q177·MathematicsSingle correctJEE Main 2021

If the value of the integral

\int_{0}^{5} \frac{x + [x]}{e^{x - [x]}} dx = \alpha e^{-1} + \beta

, where

\alpha, \beta \in \mathbf{R}

,

5\alpha + 6\beta = 0

, and

[x]

denotes the greatest integer less than or equal to x; then the value of

(\alpha + \beta)^{2}

is equal to :

(A) $100$
(B) $25$
(C) $16$
(D) $36$

Q178·MathematicsSingle correctJEE Main 2021

The value of

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{1 + \sin^{2} x}{1 + \pi^{\sin x}}\right) dx

is

(A) $\frac{\pi}{2}$
(B) $\frac{5\pi}{4}$
(C) $\frac{3\pi}{4}$
(D) $\frac{3\pi}{2}$

Q179·MathematicsSingle correctJEE Main 2021

The value of

\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}

is:

(A) $\frac{1}{2}\tan^{-1}(2)$
(B) $\frac{1}{2}\tan^{-1}(4)$
(C) $\tan^{-1}(4)$
(D) $\frac{1}{4}\tan^{-1}(4)$

Q180·MathematicsNumericalJEE Main 2021

If

\int_{0}^{\pi}\left(\sin^{3}x\right)e^{-\sin^{2}x}dx

= α −

\frac{\beta}{e}\int_{0}^{1}\sqrt{t}\ e^{t}dt

, then α + β is equal to……….

Q181·MathematicsNumericalJEE Main 2021

Let

F:\left[3,5\right] \to R

be a twice differentiable function on

\left(3,5\right)

such that

F\left(x\right)=e^{-x}\int_3^x\left(3t^2+2t+4F'\left(t\right)\right)dt

. If

F'\left(4\right)=\frac{\alpha e^{\beta}-224}{\left(e^{\beta}-4\right)^2}

, then

\alpha+\beta

is equal to ____.

Q182·MathematicsSingle correctJEE Main 2021

The value of

\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} \frac{(2j - 1) + 8n}{(2j - 1) + 4n}

is equal to :

(A) $2 - \log_e\left(\frac{2}{3}\right)$
(B) $1 + 2\log_e\left(\frac{3}{2}\right)$
(C) $5 + \log_e\left(\frac{3}{2}\right)$
(D) $3 + 2\log_e\left(\frac{2}{3}\right)$

Q183·MathematicsSingle correctJEE Main 2021

The value of the definite integral

\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{dx}{\left(1+e^{x\cos x}\right)\left(\sin^4 x+\cos^4 x\right)}

is equal to :

(A) $-\frac{\pi}{4}$
(B) $\frac{\pi}{2\sqrt{2}}$
(C) $-\frac{\pi}{2}$
(D) $\frac{\pi}{\sqrt{2}}$

Q184·MathematicsNumericalJEE Main 2021

Let the domain of the function

f\left(x\right)=\log_4\left(\log_5\left(\log_3\left(18x-x^2-77\right)\right)\right)

be

\left(a,b\right)

. Then the value of the integral

\int_a^b\frac{\sin^3 x}{\left(\sin^3 x+\sin^3\left(a+b-x\right)\right)}dx

is equal to ____.

Q185·MathematicsSingle correctJEE Main 2021

The value of the definite integral

\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan 2x}}

is :

(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{12}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{18}$

Q186·MathematicsSingle correctJEE Main 2021

The value of the integral

\int_{-1}^{1} \log\left(x + \sqrt{x^2 + 1}\right)dx

is :

(A)-1
(B)1
(C)2
(D)0

Q187·MathematicsSingle correctJEE Main 2021

If [x] denotes the greatest integer less than or equal to x, then the value of the integral

\int_{-\pi/2}^{\pi/2}\left[[x]-\sin x\right]dx

is equal to :

(A) $-\pi$
(B) $0$
(C) $1$
(D) $\pi$

Q188·MathematicsSingle correctJEE Main 2021

If

f:R\to R

is given by

f(x)=x+1,

then the value of

\lim_{n\to\infty}\frac{1}{n}\left[f(0)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+....+f\left(\frac{5(n-1)}{n}\right)\right]

, is :

(A) $\frac{3}{2}$
(B) $\frac{1}{2}$
(C) $\frac{7}{2}$
(D) $\frac{5}{2}$

Q189·MathematicsSingle correctJEE Main 2021

The value of the integral

\int_{-1}^{1} \log_e\left(\sqrt{1-x}+\sqrt{1+x}\right)dx

is equal to :

(A) $2\log_e 2+\frac{\pi}{4}-1$
(B) $\log_e 2+\frac{\pi}{2}-1$
(C) $\frac{1}{2}\log_e 2+\frac{\pi}{4}-\frac{3}{2}$
(D) $2\log_e 2+\frac{\pi}{2}-\frac{1}{2}$

Q190·MathematicsSingle correctJEE Main 2021

Let a be a positive real numbers such that

\int_{0}^{a} e^{x-[x]}dx = 10e - 9

where [x] is the greatest integer less than or equal to x. Then a is equal to :

(A) $10+\log_e 2$
(B) $10-\log_e(1+e)$
(C) $10+\log_e(1+e)$
(D) $10+\log_e 3$

Q191·MathematicsSingle correctJEE Main 2021

Let

g(t)=\int_{-\pi/2}^{\pi/2}\cos\left(\frac{\pi}{4}t+f(x)\right)dx,

where

f(x)=\log_{e}\left(x+\sqrt{x^{2}+1}\right),\ x\in R

. Then which one of the following is correct?

(A) $g(1)=g(0)$
(B) $g(1)=\sqrt{2}g(0)$
(C) $g(1)+g(0)=0$
(D) $\sqrt{2}g(1)=g(0)$

Q192·MathematicsSingle correctJEE Main 2021

Let

g(x) = \int_{0}^{x} f(t)\,dt

, where f is continuous function in [0, 3] such that

\frac{1}{3} \le f(t) \le 1

for all t ∈ [0, 1] and

0 \le f(t) \le \frac{1}{2}

for all t ∈ (1, 3]. The largest possible interval in which g(3) lies is :

(A) $\left[-1, -\frac{1}{2}\right]$
(B) $\left[-\frac{3}{2}, -1\right]$
(C) $\left[\frac{1}{3}, 2\right]$
(D)[1, 3]

Q193·MathematicsNumericalJEE Main 2021

If

[\cdot]

represents the greatest integer function, then the value of

\left| \int_{0}^{\sqrt{\frac{\pi}{2}}} \left[ \left[x^2\right] - \cos x \right] dx \right|

is _________ .

Q194·MathematicsSingle correctJEE Main 2021

Let

f : R \rightarrow R

be defined as

f(x) = e^{-x}\sin x

. If

F : [0, 1] \rightarrow R

is a differentiable function such that

F(x) = \int_{0}^{x} f(t)\,dt

, then the value of

\int_{0}^{1}\left(F'(x) + f(x)\right)e^{x}dx

lies in the interval

(A) $\left[\frac{327}{360}, \frac{329}{360}\right]$
(B) $\left[\frac{330}{360}, \frac{331}{360}\right]$
(C) $\left[\frac{331}{360}, \frac{334}{360}\right]$
(D) $\left[\frac{335}{360}, \frac{336}{360}\right]$

Q195·MathematicsSingle correctJEE Main 2021

If the integral

\int_{0}^{10}\frac{[\sin 2\pi x]}{e^{x-[x]}}dx = \alpha e^{-1} + \beta e^{-\frac{1}{2}} + \gamma

, where

\alpha, \beta, \gamma

are integers and

[x]

denotes the greatest integer less than or equal to

x

, then the value of

\alpha + \beta + \gamma

is equal to :

(A) $0$
(B) $20$
(C) $25$
(D) $10$

Q196·MathematicsNumericalJEE Main 2021

Let

I_{n} = \int_{1}^{e} x^{19}\left(\log|x|\right)^{n}dx

, where

n \in N

. If

(20)I_{10} = \alpha I_{9} + \beta I_{8}

, for natural numbers

\alpha

and

\beta

, then

\alpha - \beta

equal to _______.

Q197·MathematicsNumericalJEE Main 2021

Let f : ℝ → ℝ be a continuous function such that f(x) + f(x + 1) = 2, for all x ∈ ℝ. If

I_{1} = \int_{0}^{8} f(x)dx

and

I_{2} = \int_{-1}^{3} f(x)dx

, then the value of

I_{1} + 2I_{2}

is equal to ________ .

Q198·MathematicsNumericalJEE Main 2021

Let f : (0, 2) → ℝ be defined as

f(x) = \log_{2}\left(1+\tan\left(\frac{\pi x}{4}\right)\right)

. Then,

\lim_{n \to \infty}\frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+....+f(1)\right)

is equal to _______ .

Q199·MathematicsSingle correctJEE Main 2021

Consider the integral

I=\int\limits_{0}^{10}\frac{[x]\,e^{[x]}}{e^{x-1}}dx,

where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to:

(A) $9(e-1)$
(B) $45(e+1)$
(C) $45(e-1)$
(D) $9(e+1)$

Q200·MathematicsSingle correctJEE Main 2021

The value of

\int_{-\pi/2}^{\pi/2} \frac{\cos^{2}x}{1+3^{x}}dx

is:

(A)2π
(B)4π
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{4}$

Q201·MathematicsNumericalJEE Main 2021

The value of the integeral

\int_{0}^{\pi}\left|\sin 2x\right|\,dx

is _______

Q202·MathematicsSingle correctJEE Main 2021

For x>0, if

f(x) = \int_{1}^{x} \frac{\log_{e} t}{(1+t)} dt

, then

f(e) + f\left(\frac{1}{e}\right)

is equal to :

(A) $\frac{1}{2}$
(B)−1
(C)1
(D)0

Q203·MathematicsSingle correctJEE Main 2021

The value of

\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]}dx

, where [x] is the greatest integer ≤ x, is:

(A)100 (e–1)
(B)100e
(C)100(1-e)
(D)100 (1 + e)

Q204·MathematicsNumericalJEE Main 2021

In

I_{m,n} = \int_{0}^{1} x^{m-1}\left(1-x\right)^{n-1} dx

, for m, n

\geq

1 and

\int_{0}^{1}\frac{x^{m-1}+x^{n-1}}{\left(1+x\right)^{m+n}}dx = \alpha I_{m,n}

,

\alpha \in R

, then

\alpha

equals______________ .

Q205·MathematicsSingle correctJEE Main 2021

The value of

\int\limits_{-1}^{1} x^2 e^{[x^3]} dx

, where

[t]

denotes the greatest integer

\le t

, is :

(A) $\frac{e+1}{3}$
(B) $\frac{e-1}{3e}$
(C) $\frac{e+1}{3e}$
(D) $\frac{1}{3e}$

Q206·MathematicsNumericalJEE Main 2021

The value of ∫3x

^{2}

−3x−6 dx is ______. −2

Q207·MathematicsSingle correctJEE Main 2021

If In= ∫cot

^{n}

x dx , then: π/4

(A)I $_{2}$ 1+I $_{4}$ , I $_{3}$ 1+I $_{5}$ , I $_{4}$ 1+I $_{6}$ are in G.P.
(B)I $_{2}$ 1+I $_{4}$ , I $_{3}$ 1+I $_{5}$ , I $_{4}$ 1+I $_{6}$ are in A.P.
(C)I2 + I4, I3 + I5, I4 + I6 are in A.P.
(D)I2 + I4, (I3 + I5) $^{2}$ , I4 + I6 are in G.P.

Q208·MathematicsSingle correctJEE Main 2021

The value of the integral,

\int_{1}^{3}\left[x^{2}-2x-2\right]dx,

where [x] denotes the greatest integer less than or equal to

x

, is:

(A) $-4$
(B) $-5$
(C) $-\sqrt{2}-\sqrt{3}-1$
(D) $-\sqrt{2}-\sqrt{3}+1$

Q209·MathematicsSingle correctJEE Main 2021

Let

f(x)

be a differentiable function defined on [0,2] such that

f^{'}(x)=f^{'}(2-x)

for all

x\in(0,2)

,

f(0)=1

and

f(2)=\mathrm{e}^{2}

. Then the value of

\int_{0}^{2}f(x)dx

is:

(A) $1+e^{2}$
(B) $1-e^{2}$
(C) $2(1-e^{2})$
(D) $2(1+e^{2})$

Q210·MathematicsNumericalJEE Main 2021

If

\int_{-a}^{a} \left(|x| + |x - 2|\right) dx = 22

,

(a > 2)

and [x] denotes the greatest integer

\leq x

, then

\int_{a}^{-a} \left(x + [x]\right) dx

is equal to ______

Q211·MathematicsNumericalJEE Advanced 2020

Let

f : \mathbb{R} \to \mathbb{R}

be a differentiable function such that its derivative

f'

is continuous and

f(\pi) = -6

. If

F : [0,\pi] \to \mathbb{R}

is defined by

F(x) = \int_{0}^{x} f(t)\,dt

, and if

\int_{0}^{\pi} (f'(x) + F(x)) \cos x \, dx = 2

, then the value of

f(0)

is ________

Q212·MathematicsMultiple correctJEE Advanced 2020

Which of the following inequalities is/are TRUE?

(A) $\int_{0}^{1} x\cos x\, dx \geq \frac{3}{8}$
(B) $\int_{0}^{1} x\sin x\, dx \geq \frac{3}{10}$
(C) $\int_{0}^{1} x^{2}\cos x\, dx \geq \frac{1}{2}$
(D) $\int_{0}^{1} x^{2}\sin x\, dx \geq \frac{2}{9}$

Q213·MathematicsSingle correctJEE Main 2020

If

I_1 = \displaystyle\int_0^1 (1-x^{50})^{100}\,dx

and

I_2 = \displaystyle\int_0^1 (1-x^{50})^{101}\,dx

such that

I_2 = \alpha I_1

then

\alpha

equals to:

(A) $\dfrac{5049}{5050}$
(B) $\dfrac{5050}{5049}$
(C) $\dfrac{5050}{5051}$
(D) $\dfrac{5051}{5050}$

Q214·MathematicsSingle correctJEE Main 2020

The integral

\int_{1}^{2} e^{x} \cdot x^{2}(2+\log_{e} x)dx

equals:

(A) $e(4e + 1)$
(B) $4e^{2} - 1$
(C) $e(4e - 1)$
(D) $e(2e - 1)$

Q215·MathematicsSingle correctJEE Main 2020

The value of

\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{1}{1+e^{\sin x}}dx

is:

(A) $\dfrac{3\pi}{2}$
(B) $\dfrac{\pi}{2}$
(C) $\pi$
(D) $\dfrac{\pi}{4}$

Q216·MathematicsSingle correctJEE Main 2020

Let

f(x) = \int \frac{\sqrt{x}}{(1+x)^{2}} dx

(x \ge 0)

. Then

f(3) - f(1)

is equal to:

(A) $\frac{\pi}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}$
(B) $-\frac{\pi}{12} + \frac{1}{2} + \frac{\sqrt{3}}{4}$
(C) $\frac{\pi}{6} + \frac{1}{2} - \frac{\sqrt{3}}{4}$
(D) $-\frac{\pi}{6} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

Q217·MathematicsNumericalJEE Main 2020

Let {x} and [x] denote the fractional part of x and the greatest integer ≤x respectively of a real number x. if

\int_{0}^{n} \{x\}dx

,

\int_{0}^{n} [x]dx

and

10(n^{2} - n)

, (n ∈ N, n > 1) are three consecutive terms of a G.P. then n is equal to ……

Q218·MathematicsSingle correctJEE Main 2020

Let

f(x) = |x - 2|

and

g(x) = f(f(x))

,

x \in [0, 4]

. Then

\int\limits_{0}^{3} \left( g(x) - f(x) \right)\, dx

is equal to:

(A) $\frac{1}{2}$
(B) $\frac{3}{2}$
(C) $1$
(D) $0$

Q219·MathematicsSingle correctJEE Main 2020

The integral

\int_{\pi/6}^{\pi/3} \tan^{3} x \cdot \sin^{2} 3x \left(2\sec^{2} x \cdot \sin^{2} 3x + 3\tan x \cdot \sin 6x\right) dx

is

(A) $-\frac{1}{18}$
(B) $-\frac{1}{9}$
(C) $\frac{9}{2}$
(D) $\frac{7}{18}$

Q220·MathematicsSingle correctJEE Main 2020

\int\limits_{-\pi}^{\pi} \left| \pi - |x| \right| dx

is equal to:

(A) $\pi^{2}$
(B) $\frac{\pi^{2}}{2}$
(C) $\sqrt{2}\,\pi^{2}$
(D) $2\pi^{2}$

Q221·MathematicsSingle correctJEE Main 2020

If the value of the integral

\int_{0}^{1/2} \frac{x^2}{\left(1 - x^2\right)^{3/2}} dx

is

\frac{k}{6}

, then

k

is equal to:

(A) $2\sqrt{3} + \pi$
(B) $3\sqrt{2} - \pi$
(C) $2\sqrt{3} - \pi$
(D) $3\sqrt{2} + \pi$

Q222·MathematicsSingle correctJEE Main 2020

If for all real triplets (a, b, c),

f(x)=a+bx+cx^{2}

; then

\int_{0}^{1} f(x)\,dx

is equal to:

(A) $\dfrac{1}{2}\left\{f(1)+3f\left(\dfrac{1}{2}\right)\right\}$
(B) $\dfrac{1}{6}\left\{f(0)+f(1)+4f\left(\dfrac{1}{2}\right)\right\}$
(C) $\dfrac{1}{3}\left\{f(0)+f\left(\dfrac{1}{2}\right)\right\}$
(D) $2\left\{3f(1)+2f\left(\dfrac{1}{2}\right)\right\}$

Q223·MathematicsSingle correctJEE Main 2020

The value of

\displaystyle\int_{0}^{2\pi} \dfrac{x\sin^{8}x}{\sin^{8}x+\cos^{8}x}\,dx

is equal to:

(A) $2\pi$
(B) $4\pi$
(C) $\pi^{2}$
(D) $2\pi^{2}$

Q224·MathematicsSingle correctJEE Main 2020

If

I=\int_{1}^{2}\dfrac{dx}{\sqrt{2x^{3}-9x^{2}+12x+4}}

, then:

(A) $\dfrac{1}{16}<I^{2}<\dfrac{1}{9}$
(B) $\dfrac{1}{9}<I^{2}<\dfrac{1}{8}$
(C) $\dfrac{1}{8}<I^{2}<\dfrac{1}{4}$
(D) $\dfrac{1}{6}<I^{2}<\dfrac{1}{2}$

Q225·MathematicsSingle correctJEE Main 2020

The value of

\alpha

for which

4\alpha\int_{-1}^{2} e^{-\alpha|x|}\,dx=5

, is

(A) $\log_e\left(\frac{3}{2}\right)$
(B) $\log_e\left(\frac{4}{3}\right)$
(C) $\log_e\sqrt{2}$
(D) $\log_e 2$

Q226·MathematicsMultiple correctJEE Advanced 2019

For

a \in \mathbb{R}

|a| > 1

, let

\lim\limits_{n \to \infty}\left( \dfrac{1 + \sqrt[3]{2} + .... + \sqrt[3]{n}}{n^{7/3}\left( \dfrac{1}{(an+1)^{2}} + \dfrac{1}{(an+2)^{2}} + ... + \dfrac{1}{(an+n)^{2}} \right)} \right) = 54

Then the possible value(s) of a is/are

(A)8
(B) $-6$
(C)7
(D) $-9$

Q227·MathematicsNumericalJEE Advanced 2019

If

I = \frac{2}{\pi} \int_{-\pi/4}^{\pi/4} \frac{dx}{\left(1 + e^{\sin x}\right)\left(2 - \cos 2x\right)}

then

27 I^2

equals ____

Q228·MathematicsNumericalJEE Advanced 2019

The value of the integral

\displaystyle\int_{0}^{\pi/2} \dfrac{3\sqrt{\cos\theta}}{\left(\sqrt{\cos\theta} + \sqrt{\sin\theta}\right)^{5}}\, d\theta

equals ____

Q229·MathematicsSingle correctJEE Main 2019

If

\displaystyle\int_{0}^{\pi/2} \dfrac{\cot x}{\cot x + \text{cosec}\,x}\,dx = m(\pi + n)

, then m

\cdot

n is equal to

(A)1
(B) $\dfrac{1}{2}$
(C) $-\dfrac{1}{2}$
(D) $-1$

Q230·MathematicsSingle correctJEE Main 2019

Let f : R

\rightarrow

R be a continuously differentiable function such that f(2) = 6 and

f'(2) = \frac{1}{48}

. If

\int_{6}^{f(x)} 4t^{3}\, dt = (x-2)g(x)

, then

\lim_{x \to 2} g(x)

is equal to :

(A)24
(B)18
(C)12
(D)36

Q231·MathematicsSingle correctJEE Main 2019

A value of

\alpha

such that

\displaystyle\int_{\alpha}^{\alpha+1} \dfrac{dx}{(x+\alpha)(x+\alpha+1)} = \log_{e}\left(\dfrac{9}{8}\right)

is:

(A) $-\dfrac{1}{2}$
(B) $-2$
(C) $\dfrac{1}{2}$
(D)2

Q232·MathematicsSingle correctJEE Main 2019

The value of

\displaystyle\int_{0}^{2\pi} [\sin 2x(1+\cos 3x)]\, dx

where [t] denotes the greatest integer function, is:

(A) $\pi$
(B) $-2\pi$
(C) $2\pi$
(D) $-\pi$

Q233·MathematicsSingle correctJEE Main 2019

\displaystyle\lim_{n\to\infty}\left(\dfrac{(n+1)^{1/3}}{n^{4/3}}+\dfrac{(n+2)^{1/3}}{n^{4/3}}+....+\dfrac{(2n)^{1/3}}{n^{4/3}}\right)

is equal to:

(A) $\dfrac{3}{4}(2)^{4/3}-\dfrac{3}{4}$
(B) $\dfrac{4}{3}(2)^{3/4}$
(C) $\dfrac{3}{4}(2)^{4/3}-\dfrac{4}{3}$
(D) $\dfrac{4}{3}(2)^{4/3}$

Q234·MathematicsSingle correctJEE Main 2019

The integral

\int_{\pi/6}^{\pi/3} \sec^{2/3} x \, \operatorname{cosec}^{4/3} x \, dx

is equal to

(A) $3^{5/6} - 3^{2/3}$
(B) $3^{5/3} - 3^{1/3}$
(C) $3^{7/6} - 3^{5/6}$
(D) $3^{4/3} - 3^{1/3}$

Q235·MathematicsSingle correctJEE Main 2019

The value of

\displaystyle\int_{0}^{\pi/2} \dfrac{\sin^{3}x}{\sin x+\cos x}dx

is:

(A) $\dfrac{\pi-2}{4}$
(B) $\dfrac{\pi-1}{2}$
(C) $\dfrac{\pi-1}{4}$
(D) $\dfrac{\pi-2}{8}$

Q236·MathematicsSingle correctJEE Main 2019

The value of the integral

\displaystyle\int_{0}^{1} x\cot^{-1}\left(1-x^2+x^4\right)dx

is

(A) $\dfrac{\pi}{4}-\dfrac{1}{2}\log_e 2$
(B) $\dfrac{\pi}{2}-\log_e 2$
(C) $\dfrac{\pi}{2}-\dfrac{1}{2}\log_e 2$
(D) $\dfrac{\pi}{4}-\log_e 2$

Q237·MathematicsSingle correctJEE Main 2019

If

f(x)=\frac{2-x\cos x}{2+x\cos x}

and

g(x)=\log_{e}x,(x>0)

then the value of the integral

\displaystyle\int_{-\pi/4}^{\pi/4} g\left(f(x)\right)dx

is:

(A) $\log_{e}1$
(B) $\log_{e}2$
(C) $\log_{e}e$
(D) $\log_{e}3$

Q238·MathematicsSingle correctJEE Main 2019

Let

f(x)=\int_{0}^{x} g(t)\,dt

, were g is a non zero even function. If

f(x+5)=g(x)

, then

\int_{0}^{x} f(t)\,dt

equals

(A) $\int_{x+5}^{5} g(t)\,dt$
(B) $2\int_{5}^{x-5} g(t)\,dt$
(C) $\int_{5}^{x+5} g(t)\,dt$
(D) $5\int_{x+5}^{5} g(t)\,dt$

Q239·MathematicsSingle correctJEE Main 2019

The integral

\displaystyle\int_{1}^{e} \left\{ \left(\dfrac{x}{e}\right)^{2x} - \left(\dfrac{e}{x}\right)^{x} \right\} \log_{e} x \, dx

is equal to :

(A) $\dfrac{1}{2} - e - \dfrac{1}{e^{2}}$
(B) $-\dfrac{1}{2} + \dfrac{1}{e} - \dfrac{1}{2e^{2}}$
(C) $\dfrac{3}{2} - \dfrac{1}{e} - \dfrac{1}{2e^{2}}$
(D) $\dfrac{3}{2} - e - \dfrac{1}{2e^{2}}$

Q240·MathematicsSingle correctJEE Main 2019

\lim_{x\to\infty} \left( \dfrac{n}{n^{2}+1^{2}} + \dfrac{n}{n^{2}+2^{2}} + \dfrac{n}{n^{2}+3^{2}} + \ldots + \dfrac{1}{5n} \right)

is equal to

(A) $\dfrac{\pi}{4}$
(B) $\tan^{-1}(3)$
(C) $\dfrac{\pi}{2}$
(D) $\tan^{-1}(2)$

Q241·MathematicsSingle correctJEE Main 2019

Let f and g be continuous functions on

[0, a]

such that

f(x)=f(a-x)

and

g(x)+g(a-x)=4

, then

\int_{0}^{a}f(x)g(x)dx

is equal to:

(A) $4\int_{0}^{a}f(x)dx$
(B) $\int_{0}^{a}f(x)dx$
(C) $2\int_{0}^{a}f(x)dx$
(D) $-3\int_{0}^{a}f(x)dx$

Q242·MathematicsSingle correctJEE Main 2019

The value of the integral

\int_{-2}^{2}\dfrac{\sin^{2}x}{\left[\dfrac{x}{\pi}\right]+\dfrac{1}{2}}\,dx

(where [x] denotes the greatest integer less than or equal to x) is:

(A)0
(B) $\sin 4$
(C)4
(D) $4-\sin 4$

Q243·MathematicsSingle correctJEE Main 2019

Let

I = \displaystyle\int_{a}^{b} (x^4 - 2x^2)\,dx

. If I is minimum then the ordered pair (a, b) is:

(A) $(0, \sqrt{2})$
(B) $(-\sqrt{2}, 0)$
(C) $(\sqrt{2}, -\sqrt{2})$
(D) $(-\sqrt{2}, \sqrt{2})$

Q244·MathematicsSingle correctJEE Main 2019

The value of

\displaystyle\int_{0}^{\pi} |\cos x|^3\, dx

is:

(A)0
(B) $\dfrac{4}{3}$
(C) $\dfrac{2}{3}$
(D) $-\dfrac{4}{3}$

Q245·MathematicsSingle correctJEE Main 2019

If

\int_{0}^{\frac{\pi}{3}} \frac{\tan\theta}{\sqrt{2k\sec\theta}} d\theta = 1 - \frac{1}{\sqrt{2}}, (k > 0)

, then the value of k is:

(A) $2$
(B) $\frac{1}{2}$
(C) $4$
(D) $1$

Q246·MathematicsNumericalJEE Advanced 2018

The value of the integral

\int_{0}^{1/2} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{1/4}}\, dx

is ______ .

Q247·MathematicsMultiple correctJEE Advanced 2017

Let

f : R \to (0, 1)

be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval

(0, 1)

?

(A) $e^{x} - \int_{0}^{x} f(t)\sin t\, dt$
(B) $x^{9} - f(x)$
(C) $f(x) + \int_{0}^{\pi/2} f(t)\sin t\, dt$
(D) $x - \int_{0}^{\frac{\pi}{2} - x} f(t)\cos t\, dt$

Q248·MathematicsMultiple correctJEE Advanced 2017

If

I = \displaystyle\sum_{k=1}^{98} \int_{k}^{k+1} \dfrac{k+1}{x(x+1)}\,dx

, then

(A) $I > \log_{e} 99$
(B) $I < \log_{e} 99$
(C) $I < \dfrac{49}{50}$
(D) $I > \dfrac{49}{50}$

Q249·MathematicsIntegerJEE Advanced 2017

Let

f : R \to R

be a differentiable function such that

f(0) = 0

,

f\left(\frac{\pi}{2}\right) = 3

and

f'(0) = 1

. If

g(x) = \int_{x}^{\pi/2} \left[ f'(t)\operatorname{cosec} t - \cot t\ \operatorname{cosec} t\ f(t) \right] dt

for

x \in \left(0, \frac{\pi}{2}\right]

, then

\lim_{x \to 0} g(x) =

Q250·MathematicsIntegerJEE Advanced 2016

The total number of distinct

x \in [0, 1]

for which

\int_{0}^{x} \frac{t^{2}}{1 + t^{4}}\,dt = 2x - 1

is

Q251·MathematicsSingle correctJEE Advanced 2016

The value of

\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{2} \cos x}{1 + e^{x}} \, dx

is equal to

(A) $\frac{\pi^{2}}{4} - 2$
(B) $\frac{\pi^{2}}{4} + 2$
(C) $\pi^{2} - e^{\frac{\pi}{2}}$
(D) $\pi^{2} + e^{\frac{\pi}{2}}$

Q252·MathematicsIntegerJEE Advanced 2015

Let

f : \mathbb{R} \to \mathbb{R}

be a function defined by

f(x) = \begin{cases} [x], & x \le 2 \\ 0, & x > 2 \end{cases}

, where

[x]

is the greatest integer less than or equal to

x

. If

I = \displaystyle\int_{-1}^{2} \dfrac{x f(x^{2})}{2 + f(x+1)}\, dx

, then the value of

(4I - 1)

is

Q253·MathematicsIntegerJEE Advanced 2015

Let

f : \mathbb{R} \to \mathbb{R}

be a continuous odd function, which vanishes exactly at one point and

f(1) = \dfrac{1}{2}

. Suppose that

F(x) = \displaystyle\int_{-1}^{x} f(t)\,dt

for all

x \in [-1, 2]

and

G(x) = \displaystyle\int_{-1}^{x} t\left|f\left(f(t)\right)\right|dt

for all

x \in [-1, 2]

. If

\displaystyle\lim_{x \to 1}\dfrac{F(x)}{G(x)} = \dfrac{1}{14}

, then the value of

f\left(\dfrac{1}{2}\right)

is

Q254·MathematicsMultiple correctJEE Advanced 2015

The option(s) with the values of

a

and

L

that satisfy the following equation is(are)

\dfrac{\displaystyle\int_{0}^{4\pi} e^{t}\left(\sin^{6}at + \cos^{4}at\right)dt}{\displaystyle\int_{0}^{\pi} e^{t}\left(\sin^{6}at + \cos^{4}at\right)dt} = L

?

(A) $a = 2$ , $L = \dfrac{e^{4\pi} - 1}{e^{\pi} - 1}$
(B) $a = 2$ , $L = \dfrac{e^{4\pi} + 1}{e^{\pi} + 1}$
(C) $a = 4$ , $L = \dfrac{e^{4\pi} - 1}{e^{\pi} - 1}$
(D) $a = 4$ , $L = \dfrac{e^{4\pi} + 1}{e^{\pi} + 1}$

Q255·MathematicsMultiple correctJEE Advanced 2015

Let

f'(x) = \dfrac{192x^{3}}{2 + \sin^{4}\pi x}

for all

x \in \mathbb{R}

with

f\left(\dfrac{1}{2}\right) = 0

. If

m \le \displaystyle\int_{1/2}^{1} f(x)\,dx \le M

, then the possible values of

m

and

M

are

(A) $m = 13$ , $M = 24$
(B) $m = \dfrac{1}{4}$ , $M = \dfrac{1}{2}$
(C) $m = -11$ , $M = 0$
(D) $m = 1$ , $M = 12$

Q256·MathematicsMultiple correctJEE Advanced 2015

Let

F : \mathbb{R} \to \mathbb{R}

be a thrice differentiable function. Suppose that

F(1) = 0

,

F(3) = -4

and

F'(x) < 0

for all

x \in (1/2, 3)

. Let

f(x) = xF(x)

for all

x \in \mathbb{R}

. If

\displaystyle\int_{1}^{3} x^{2}F'(x)\,dx = -12

and

\displaystyle\int_{1}^{3} x^{3}F''(x)\,dx = 40

, then the correct expression(s) is(are)

(A) $9f'(3) + f'(1) - 32 = 0$
(B) $\displaystyle\int_{1}^{3} f(x)\,dx = 12$
(C) $9f'(3) - f'(1) + 32 = 0$
(D) $\displaystyle\int_{1}^{3} f(x)\,dx = -12$

Q257·MathematicsIntegerJEE Advanced 2015

Let

F(x) = \displaystyle\int_{x}^{x^{2}+\frac{\pi}{6}} 2\cos^{2} t\, dt

for all

x \in \mathbb{R}

and

f : \left[0, \dfrac{1}{2}\right] \to [0, \infty)

be a continuous function. For

a \in \left[0, \dfrac{1}{2}\right]

, if

F'(a) + 2

is the area of the region bounded by

x = 0

,

y = 0

,

y = f(x)

and

x = a

, then

f(0)

is

Q258·MathematicsMultiple correctJEE Advanced 2015

Let

f(x) = 7\tan^{8}x + 7\tan^{6}x - 3\tan^{4}x - 3\tan^{2}x

for all

x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)

. Then the correct expression(s) is(are)

(A) $\displaystyle\int_{0}^{\pi/4} x f(x)\,dx = \dfrac{1}{12}$
(B) $\displaystyle\int_{0}^{\pi/4} f(x)\,dx = 0$
(C) $\displaystyle\int_{0}^{\pi/4} x f(x)\,dx = \dfrac{1}{6}$
(D) $\displaystyle\int_{0}^{\pi/4} f(x)\,dx = 1$

Q259·MathematicsIntegerJEE Advanced 2015

If

\alpha = \displaystyle\int_{0}^{1}\left(e^{9x + 3\tan^{-1}x}\right)\left(\dfrac{12 + 9x^{2}}{1 + x^{2}}\right)dx

where

\tan^{-1}x

takes only principal values, then the value of

\left(\log_{e}\left|1 + \alpha\right| - \dfrac{3\pi}{4}\right)

is

Q260·MathematicsMultiple correctJEE Advanced 2014

Let

f: (0, \infty) \to \mathbb{R}

be given by

f(x) = \int_{1/x}^{x} e^{-\left(t+\frac{1}{t}\right)} \frac{dt}{t}

, then

(A) $f(x)$ is monotonically increasing on $[1, \infty)$
(B) $f(x)$ is monotonically decreasing on $(0, 1)$
(C) $f(x) + f\left(\frac{1}{x}\right) = 0$ , for all $x \in (0, \infty)$
(D) $f(2^x)$ is an odd function of $x$ on $\mathbb{R}$

Q261·MathematicsSingle correctJEE Advanced 2014

Match the following:

List – I	List – II
P.The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\le 2$ , satisfying $f(0) = 0$ and $\int_{0}^{1} f(x)\, dx = 1$ , is	1.8
Q.The number of points in the interval $\left[-\sqrt{13}, \sqrt{13}\right]$ at which $f(x) = \sin(x^{2}) + \cos(x^{2})$ attains its maximum value, is	2.2
R. $\int_{-2}^{2} \frac{3x^{2}}{(1+e^{x})}\, dx$ equals	3.4
S. $\frac{\left(\int_{-1/2}^{1/2} \cos 2x \cdot \log\left(\frac{1+x}{1-x}\right) dx\right)}{\left(\int_{0}^{1/2} \cos 2x \cdot \log\left(\frac{1+x}{1-x}\right) dx\right)}$ equals	4.0

(A)P-3, Q-2, R-4, S-1
(B)P-2, Q-3, R-4, S-1
(C)P-3, Q-2, R-1, S-4
(D)P-2, Q-3, R-1, S-4

Q262·MathematicsSingle correctJEE Advanced 2014

Given that for each

a \in (0, 1)

,

\lim_{h \to 0^{+}} \int_{h}^{1-h} t^{-a} (1-t)^{a-1}\, dt

exists. Let this limit be

g(a)

. In addition, it is given that the function

g(a)

is differentiable on

(0, 1)

. The value of

g\left(\frac{1}{2}\right)

is

(A) $\pi$
(B) $2\pi$
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{4}$

Q263·MathematicsSingle correctJEE Advanced 2014

Let

f : [0, 2] \to \mathbb{R}

be a function which is continuous on

[0, 2]

and is differentiable on

(0, 2)

with

f(0) = 1

. Let

F(x) = \int_{0}^{x^{2}} f\left(\sqrt{t}\right) dt

for

x \in [0, 2]

. If

F'(x) = f'(x)

for all

x \in (0, 2)

, then

F(2)

equals

(A) $e^{2} - 1$
(B) $e^{4} - 1$
(C) $e - 1$
(D) $e^{4}$

Q264·MathematicsIntegerJEE Advanced 2014

The value of

\int_{0}^{1} 4x^3 \left\{\frac{d^2}{dx^2}(1 - x^2)^5\right\} dx

is __________

Q265·MathematicsSingle correctJEE Advanced 2014

Given that for each

a \in (0, 1)

,

\lim_{h \to 0^{+}} \int_{h}^{1-h} t^{-a} (1-t)^{a-1}\, dt

exists. Let this limit be

g(a)

. In addition, it is given that the function

g(a)

is differentiable on

(0, 1)

. The value of

g'\left(\frac{1}{2}\right)

is

(A) $\frac{\pi}{2}$
(B) $\pi$
(C) $-\frac{\pi}{2}$
(D)0

Q266·MathematicsSingle correctJEE Advanced 2014

The following integral

\int_{\pi/4}^{\pi/2} (2\operatorname{cosec} x)^{17}\, dx

is equal to

(A) $\int_{0}^{\log(1+\sqrt{2})} 2\left(e^{u} + e^{-u}\right)^{16}\, du$
(B) $\int_{0}^{\log(1+\sqrt{2})} \left(e^{u} + e^{-u}\right)^{17}\, du$
(C) $\int_{0}^{\log(1+\sqrt{2})} \left(e^{u} - e^{-u}\right)^{17}\, du$
(D) $\int_{0}^{\log(1+\sqrt{2})} 2\left(e^{u} - e^{-u}\right)^{16}\, du$

Q267·MathematicsSingle correctJEE Advanced 2013

Let

f:[\frac{1}{2},1]\to R

(the set of all real numbers) be a positive, non-constant and differentiable function such that

f'(x)<2f(x)

and

f(\frac{1}{2})=1

. Then the value of

\int_{1/2}^{1}f(x)\,dx

lies in the interval

(A) $(2e-1,2e)$
(B) $(e-1,2e-1)$
(C) $(\frac{e-1}{2},e-1)$
(D) $(0,\frac{e-1}{2})$

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